DE102012000044A1 - Method for interaction-free entanglement of quantum system in quantum computer, involves locating quantum system by activating magnetic field, and transferring quantum system to entangled condition under observation of boundary condition - Google Patents

Abstract

The method involves preparing a quantum system as an elementary quantum system in a condition with real phases. The quantum system is located by activating homogeneous magnetic field comprising domains. The quantum system is transferred to an entangled condition under observation of boundary condition, where the quantum system is realized by 13 carbon-atoms in diamond. The quantum system is addressed and/or read by nitrogen-vacancy center, and the magnetic field is produced by low-inductive coils in the form of a Helmholtz arrangement.

Description

The invention relates to a method for the interaction-free entanglement of quantum systems in quantum computers.

For entanglement of quantum systems in quantum computers it is known that the quantum systems to be entangled can be prepared in the desired entangled state by means of physical interactions, such as Coulomb repulsion. For example, in the method described in [17], ions trapped in a linear ion trap can be prepared in an entangled state due to the Coulomb repulsion occurring between the ions.

A major disadvantage of this type of state preparation can be seen in the fact that the involved quantum systems must interact with each other for this purpose. Since, however, it is not possible to accommodate any number of ions in a linear ion trap, for example, a plurality of ion traps are required to construct a quantum computer. This means, however, that also between these appropriate preparation steps must be possible. It would therefore be desirable to have a method that allows interaction-free entanglement of quantum systems. A corresponding method for the interaction-free entanglement of quantum systems in different ion traps is described in [53]. The method provides the following steps: In order to convert the previously suitably prepared quantum systems into an entangled state, they are optically excited by means of ultrashort laser pulses. The excited quantum systems then spontaneously emit one photon each. These are then superimposed on a beam splitter and then detected. Only π-polarized photons are considered. If a photon is detected at both outputs of the beam splitter at the same time, it can be concluded that the two quantum systems were successfully crossed. The main disadvantage of this method, however, can be seen in the fact that a classical information channel is needed in this method. Another significant disadvantage of this method can be seen in the fact that the probability of a successful entanglement is <3 × 10 ^-8 and the required apparatus is quite considerable. Also, it would be very expensive to increase the probability that it is acceptable for practical applications.

Proceeding from this, the object of the invention is to develop a method for entangling quantum systems in quantum computers, which requires neither an interaction nor a classical information channel. Only then can a simple and cost-effective entanglement of quantum systems be made possible, even if they are far from one another. The invention is based on the finding that not only entangled quantum systems possess properties which can only be understood as a property of the overall system, but not as the sum of the properties of the subsystems, but also possess certain product states properties which are only a characteristic of the overall system, but can not be construed as the sum of the properties of the subsystems. Based on this knowledge, a method for the interaction-free entanglement of quantum systems is then developed.

To solve this problem, the feature combinations specified in the method claims 1 to 7 are proposed. Advantageous embodiments and modifications of the invention will become apparent from the dependent claims.

Although the invention can be formally described in the context of quantum physics, it is necessary for an understanding of the invention to be able to interpret the formalism used adequately. Since no truly satisfactory interpretation can be given for quantum physics until today, part of the invention deals with the development of an interpretation for quantum physics. The starting point for the interpretation approach proposed here is the interpretation approach already presented in [49]. I would like to repeat this here and expand and deepen it in the points essential to the invention. For quantum physics as a whole, however, the interpretation proposed here can only be understood as an approach.

• I. Einleitung: Ich möchte hier kurz die grundsätzlichen Überlegungen zusammenfassen und die diesen zugrundeliegenden Vorstellungen erläutern.
• II. Ein erster Interpretationsansatz: Um zu einer befriedigenden Interpretation für die Quantenphysik gelangen zu können, ist es erforderlich, ein Verständnis für die zentralen Begriffe zu entwickeln. Hierzu werde ich aufzeigen, dass die betrachteten Quantensysteme, um im Rahmen des in der Quantenphysik postulierten Superpositionsprinzips überhaupt von prinzipiell ununterscheidbaren Möglichkeiten sprechen zu können, eine unverwechselbare Eigenschaft besitzen müssen. Diese unverwechselbare Eigenschaft ist für die hier betrachteten Quantensysteme immer über die Energie gegeben, die den Quantensystemen zugeordnet werden kann. Die, einem Quantensystem zugeordnete Energie ermöglicht es dann, diesem eine energetische Repräsentation zuzuordnen. Es lässt sich dann zeigen, dass Informationen über den Zustand eines Quantensystems nur über die für das betrachtete Quantensystem möglichen energetischen Repräsentationen zugänglich sind.
• III. Eine energetische Betrachtung einfacher Modellsysteme: Den im II. Abschnitt eingeführten Interpretationsansatz möchte ich hier vertiefen und weiter ausführen. Die betrachteten Quantensysteme werden durch ⁴⁰Ca⁺-Ionen gebildet, die sich in einer linearen Ionenfalle befinden. Ich möchte auch kurz auf das Messproblem eingehen und aufzeigen, dass dieses als unmittelbare Folge des Superpositionsprinzips aufgefasst werden kann. Für die hier betrachteten Modellsysteme kann man dann den tieferen Grund dafür, dass einzelne Messwerte prinzipiell nicht vorhergesagt werden können, wenn mehrere Messwerte möglich sind, auch darin sehen, dass die in den einzelnen Ionen gespeicherte Energie quantisiert ist. Weiter werde ich aufzeigen, dass die über die Rechenbasis zugängliche Information über ein Quantensystem, mit der Information übereinstimmt, die über die, für dieses Quantensystem möglichen energetischen Repräsentationen, zugänglich ist. Ich werde auch aufzeigen, dass man über den Zustand einem Quantensystem innere Eigenschaften zuordnen kann. Können den inneren Eigenschaften die entsprechenden energetischen Repräsentationen zugeordnet werden, so können diese als physikalisch realisiert angesehen werden. Einem Quantensystem können aber auch inhärente Eigenschaften zugeordnet werden. Diese inneren Eigenschaften zeichnen sich dadurch aus, dass diesen nicht die entsprechenden energetischen Repräsentationen zugeordnet werden können. Allerdings ist es möglich, formal äquivalente Quantensysteme einzuführen, über die die inhärenten Eigenschaften eines Quantensystems dann zugänglich werden. Ein bemerkenswertes Ergebnis dieser Überlegungen ist, dass die inneren Eigenschaften bestimmter Produktzustände nur noch als physikalisch realisierte Eigenschaft des Gesamtsystems aufgefasst werden können. Basierend auf diesen Überlegungen werde ich dann einen Vorschlag für eine die Quantenphysik auszeichnende Idee vorstellen.
• IV. Elementare Quantensysteme: Nach den im III. Abschnitt angestellten Überlegungen könnte es auch Quantensysteme geben, denen man keine energetische Repräsentation mehr zuordnen kann. Das es solche Systeme gibt, werde ich anhand von Spin ½ Systemen aufzeigen und die für die weitere Diskussion wesentlichen Eigenschaften erläutern. Eine wesentliche Eigenschaft der hier betrachteten elementaren Quantensysteme besteht darin, dass diese jederzeit mittels eines geeignet gewählten Präparationsschrittes in ein energetisch reprasentiertes Quantensystem überführt werden können. Auch ist immer der umgekehrte Präparationsschritt möglich. Hierzu müssen einzelne Spin ½ Systeme allerdings offensichtlich mit der Umgebung physikalisch in Wechselwirkung treten.
• V. Kohärent gekoppelte Vakuumfluktuationen: Überlegungen zum Casimir-Effekt, legen jedoch die Vermutung nahe, dass bestimmte Quantensysteme, die aus mindestens zwei Spin ½ Systemen zusammengesetzt sind, auch ohne eine physikalische Wechselwirkung mit der Umgebung und ohne eine physikalische Wechselwirkung untereinander, in die entsprechenden energetisch repräsentierten Quantensysteme überführt werden könnten. Der hier postulierte Prozess der kohärent gekoppelten Vakuumfluktuation könnte diesen Präparationsschritt ermöglichen. Der Prozess der kohärent gekoppelten Vakuumfluktuation würde dann zu einem simultanen Umklappen der Spinzustände der beteiligten Spin ½ Systeme führen.
• VI. Ein Gedankenexperiment zur wechselwirkungsfreien Informationsübertragung: Das hier vorgeschlagene Gedankenexperiment bietet die Möglichkeit, den im V. Abschnitt postulierten Prozess der kohärent gekoppelten Vakuumfluktuation überprüfen zu können. Wenn die postulierten kohärent gekoppelten Vakuumfluktuationen auftreten können, so besteht die einzig mögliche Interpretation dieses Experimentes darin, dass die physikalische Welt nicht kausal abgeschlossen sein kann.
• VII: Einbindung des Spin-Statistik-Therorems in den Interpretationsansatz. Ich möchte hier der Frage nachgehen, welche Konsequenzen sich für den in dieser Arbeit entwickelten Interpretationsansatz aus dem in der Quantenphysik gültigen Spin-Statistik-Theorem ergeben. Eine sich hieraus ergebende Konsequenz ist, dass bestimmte Produktzustände mit physikalisch realisierten inneren Eigenschaften eine wechselwirkungsfrei Verschränkung von Quantensystemen ermöglichen sollten. Zur Überprüfung dieser Schlussfolgerung werde ich ein einfaches Gedankenexperiment vorschlagen.
• VIII. Philosophische Konsequenzen: Ich möchte hier noch kurz auf einige philosophische Konsequenzen eingehen, die weit über die, für unser physikalisches Weltbild relevanten Fragen hinausgehen. Auch möchte ich auf mögliche Zusammenhänge hinweisen, die zwischen den bisher vorgetragenen Überlegungen und den Möglichkeiten des menschlichen Bewusstseins bestehen könnten.

• I. Einleitung: Ich möchte hier kurz die grundsätzlichen Überlegungen zusammenfassen und die diesen zugrundeliegenden Vorstellungen erläutern.

The description of the invention is therefore subdivided into the following sections:

• I. Introduction: I would like to briefly summarize the fundamental considerations and explain the underlying ideas.
• II. A First Interpretation Approach: In order to arrive at a satisfactory interpretation for quantum physics, it is necessary to develop an understanding of the central concepts. For this I will show that the considered quantum systems, in the context of the postulated in quantum physics Superposition principle to be able to speak of fundamentally indistinguishable possibilities, must have a distinctive property. For the quantum systems under consideration, this unique property is always given by the energy that can be assigned to the quantum systems. The energy associated with a quantum system then makes it possible to assign an energy representation to it. It can then be shown that information about the state of a quantum system is only accessible via the energy representations possible for the quantum system under consideration.
• III. An energetic consideration of simple model systems: I would like to deepen and further elaborate the interpretation approach introduced in Section II. The considered quantum systems are formed by ⁴⁰ Ca ⁺ ions, which are located in a linear ion trap. I would also like to deal briefly with the measurement problem and show that this can be understood as a direct consequence of the superposition principle. For the model systems considered here, one can then see the deeper reason that individual measured values can not be predicted in principle, if several measured values are possible, also in the fact that the energy stored in the individual ions is quantized. Furthermore, I will show that the information accessible via the computational base about a quantum system agrees with the information accessible via the energetic representations possible for this quantum system. I will also show that one can attribute internal properties to a quantum system via the state. If the corresponding energy representations can be assigned to the inner properties, then these can be regarded as physically realized. However, inherent properties can also be assigned to a quantum system. These inner qualities are characterized by the fact that they can not be assigned the corresponding energetic representations. However, it is possible to introduce formally equivalent quantum systems, which then make the inherent properties of a quantum system accessible. A remarkable result of these considerations is that the intrinsic properties of certain product states can only be understood as a physically realized property of the whole system. Based on these considerations, I will then present a proposal for quantifying the idea of quantum physics.
• IV. Elementary Quantum Systems: According to the III. There might also be quantum systems to which no energetic representation can be assigned. The existence of such systems will be demonstrated using spin ½ systems and will explain the essential features for further discussion. An essential feature of the elementary quantum systems considered here is that they can be converted at any time by means of a suitably selected preparation step into an energetically represented quantum system. Also, the reverse preparation step is always possible. However, individual spin ½ systems obviously have to physically interact with the environment.
• V. Coherently Coupled Vacuum Fluctuations: Considerations on the Casimir effect suggest, however, that certain quantum systems composed of at least two spin ½ systems, even without a physical interaction with the environment and without a physical interaction with each other, into the corresponding energetically represented quantum systems could be transferred. The process of coherently coupled vacuum fluctuation postulated here could make this preparation step possible. The process of coherently coupled vacuum fluctuation would then lead to a simultaneous collapse of the spin states of the participating spin ½ systems.
• VI. A thought experiment on the interaction-free transfer of information: The thought experiment proposed here offers the opportunity to be able to check the process of coherently coupled vacuum fluctuation postulated in the fifth section. If the postulated coherent coupled vacuum fluctuations can occur, the only possible interpretation of this experiment is that the physical world can not be causally closed.
• VII: Integration of the Spin Statistics Theremem into the Interpretation Approach. I would like to investigate the question which consequences arise for the interpretation approach developed in this work from the spin-statistics theorem valid in quantum physics. A consequence of this is that certain product states with physically realized internal properties should allow an interaction-free entanglement of quantum systems. To test this conclusion, I will propose a simple thought experiment.
• VIII. Philosophical Consequences: I would like to briefly discuss here some philosophical consequences that go far beyond the questions relevant to our physical world view. I would also like to point out possible connections that could exist between the considerations presented so far and the possibilities of human consciousness.

• I. Introduction: I would like to briefly summarize the fundamental considerations and explain the underlying ideas.

At the heart of this work is the question of whether two physically independent, spatially separated quantum systems, between which there is no interaction, can be converted into an entangled state even if no classical information channel is available.

In all the examples so far considered in the physical literature, either an interaction or a classical information channel is always required in order to be able to convert two spatially separated, independent quantum systems into an entangled state.

For example, in the experiments described in [17], the ⁴⁰ Ca ⁺ ions stored in the linear ion trap must be able to interact via Coulomb repulsion in order to convert them into an entangled state starting from a product state. However, it is not possible in this way to entangle any desired number of ions, since the degree of freedom "of the common vibration of all ions along the longitudinal axis of the ion trap" required for the preparation process can no longer be sufficiently selectively excited or excited with increasing ion number.

In order to be able to realize a quantum computer, it would therefore be desirable to have a preparation method at hand, which makes it possible to convert two independent ions into an entangled state even if they are located in different, spatially separated ion traps.

The experiments described in [53] are a first step in this direction. For this purpose, in each case a Yb ⁺ ion is captured in two spatially separated ion traps. These are prepared in a first step in a suitable product state. In order to be able to convert the ions thus prepared into the desired entangled state, both ions are simultaneously optically excited by means of suitable ultrashort laser pulses. The excited ions then spontaneously emit one photon each. These are then superimposed on a beam splitter and then detected. Only π-polarized photons are considered. If a photon is detected at the two outputs of the beam splitter at the same time, it can be concluded that the two ions were successfully entangled into the desired quantum state. However, the probability of successful entanglement in these experiments is only <3 × 10 ^-8 .

Although the experiments described in [53] involve an interaction-free entanglement of the two quantum systems, a classical information channel is also required in these experiments in order to successfully transform the two quantum systems into the desired entangled state.

In order to understand the basic idea of the method proposed in this work for the interaction-free entanglement of spatially separated quantum systems in which no classical information channel is required, it is helpful to take a closer look at the formalism of quantum physics.

In contrast to classical theories, such as classical mechanics but also the special theory of relativity, in which the respective formalism can be understood as a direct description of physical events in the physical space, has been in the quantum physics so far failed to give a plausible idea how the quantities occurring in the formalism of quantum physics can be assigned to physical events. In the context of quantum physics, the properties assigned to a quantum system are described by the state determined via the respective preparation process. For the development of time (state dynamics), quantum physics, for a closed quantum system, requires that it must be describable by means of unitary operators. By means of this unitary state dynamics, however, it is not possible to describe a measurement process. What characterizes a measurement process and how it can be correctly formally described is, strictly speaking, still an unsolved problem. Within the framework of the standard interpretation the known measurement problem. Also, the decoherence theory in which the environment, in particular the measuring device, is included in the quantum physical description can not answer this question.

However, it is at least clear that the state assigned to a quantum system can not be understood as an immediate description of events in physical space. But then how can the collapse of the state postulated in the framework of the standard interpretation via the projection postulate be physically interpreted during a measurement and which physical processes occur in this process?

A very simple and consistent description of these processes allows the interpretation approach presented in Sections II and III. In essence, this starts from the following idea:
In nature, there are properties that are in principle not accessible via physical interactions. One could call these properties internal properties of quantum systems. The intrinsic properties physically attributable to a quantum system are determined by the state that is assigned to the quantum system under consideration in quantum physics. However, the mathematical object "state" not only provides a description of the intrinsic properties of a quantum system and its unitary state dynamics, but also determines how the intrinsic properties of a quantum system are correlated with the energetic properties of the quantum system. There is a reciprocal, but not a one-to-one relationship between the intrinsic properties of a quantum system and its energetic properties. On the one hand, the inner properties that can be physically attributed to a quantum system clearly define the energetic properties that are possible for the quantum system (and thus the possible measurements on the quantum system). On the other hand, the possible or actually realized energetic properties of the quantum system neither clearly nor completely determine the intrinsic properties physically attributable to the quantum system. It just seems as though the knowledge gained through a preparation process, but also via a measurement process, via a quantum system forces a coupling between the intrinsic properties of the quantum system and the corresponding energetic properties of the quantum system. A preparation process differs from a measurement process in that during a preparation process one of the possible energetic properties of the quantum system can actually be realized, while in a measurement process only knowledge arises about which energetic properties can be assigned to a quantum system, whereby the actually realized energetic property of the quantum system However, the quantum system is not changed during the measurement (this of course only applies if the quantum system is not destroyed during the measurement).

The easiest way to illustrate this idea is by means of an example. In the linear ion trap described in [17], two ⁴⁰ Ca ⁺ ions enter the singlet state Ψ _- = 1/2 ^1/2 (| 1> ₂ | 0)> ₁ - | 0> ₂ | 1> ₁ ) prepared, this is clearly defined except for a global unobservable phase. When measuring on one of the two ions, two measurement results are possible. "Ion fluoresces", then the ion is present in the ground state | 1> after the measurement and "ion does not fluoresce", then the ion is present in the excited state | 0> after the measurement (see ). However, for an ion to be in the excited state after the measurement result "ion does not fluoresce", the ion must have the energy hf ₀₁ (h denotes the Plank effect quantum and f ₀₁ the transition from | 1> to | 0 > resonant stimulating transition frequency). Therefore, if one prepares the ion from the ground state (which is assigned the energy zero) by a preparation step into an arbitrary state and then obtains the measurement result "ion does not fluoresce", the ion must have absorbed the energy hf ₀₁ at some point during the preparation step. since only the manipulation laser during the preparation step can provide light quanta with the energy hf _d1 . However, the following energetic properties can be assigned to the two ions in the singlet state: 1.) "In 50% of the cases, the ion 1 has the energy hf ₀₁ and the ion 2 the energy zero" and 2.) "In 50 % of cases ion 2 has energy hf ₀₁ and ion 1 energy zero ". In principle, it is not possible to decide which case actually exists. Since the energy stored in the individual ions clearly determines the measurement result even before the measurement, it can then be said that the energy stored in the ions represents the state "energetically". The energetic representations possible for the singlet state are then: 1.) "The ion 1 has stored the energy hf ₀₁ and the ion 2 has no energy stored" and 2.) "The ion 2 has stored the energy hf ₀₁ and that Ion 1 has no energy stored ". Thus, a measurement can only change our knowledge of the energetic properties attributed to a quantum system. However, the actually realized energetic properties (the actually realized energetic representation) of the quantum system are not changed by a measurement.

If, for example, the result obtained on Ion 1 is "No fluorescence of ions", then one knows that the quantum system has realized the energetic representation "Ion 1 has stored the energy hf ₀₁ and the ion 2 has not stored any energy". The result of this knowledge is that the intrinsic properties associated with the singlet state can no longer be coupled to the energetic properties of the quantum system, since the knowledge of the quantum system obtained by the measurement implies a coupling of the states associated with the state | forces internal properties on the energetic properties of the quantum system. It does not matter if this knowledge actually exists. It is only essential that it is possible in principle that this knowledge could at any time be assigned to the two ions. Inner qualities can neither be created nor destroyed in this sense. You can only have an effect on the considered quantum system or not. Depending on the knowledge about the quantum system.

If this idea is correct, then the question arises as to whether a quantum system must necessarily realize an energetic representation. That this is not the case is shown in Section IV using spin 1/2 systems. If, for example, two ⁴⁰ Ca ⁺ ions are prepared in the singlet state 1/2 ^1/2 (| 1> ₂ | 2> ₁ - | 2> ₂ | 1> ₁ ) in the linear ion trap described in [17], this results in the following situation: Both ions are present in the electronic ground state (4 ² S _1/2 ). Since the nuclear spin of the ⁴⁰ Ca ⁺ ions used equals zero, the state 1/2 ^1/2 (| 1> ₂ | 2> ₁ - | 2> ₂ | 1> ₁ ) describes a system composed of two spin ½ systems with total spin 0. The electronic ground state is transformed by the Zeemann effect into the energetically lower state | 1> with the magnetic quantum number m = -1/2 (this is assigned the energy -ΔE _z / 2) and the higher energy state | 2> with the magnetic quantum number m = +1/2 (this is assigned to the energy + ΔE _z / 2). The quantum system can thus be assigned energy properties via the interaction energy, which energetically represent the physical properties attributable to the quantum system (ΔE _z > 0 denotes the energy difference between the two states | 2> and | 1> with the magnetic field switched on). If the magnetic field superimposed in the z-direction of the ion trap is switched off, the two energy levels | 1> and | 2> coincide. As long as the magnetic field is switched off, the system can not be assigned an energetic representation in the state 1/2 ^1/2 (| 1> ₂ | 2> ₁ - | 2 ₂ | 1> ₁ ) since the two energy levels | 1> and | 2> then coincide energetically and thus it is not possible for the quantum system to realize the energetic properties corresponding to the physical properties physically attributable to the quantum system. Quantum systems that have no energetic representations could be called "elementary quantum systems". By means of the spin degree of freedom it is thus possible to prepare elementary quantum systems. If the magnetic field is switched on again, this automatically generates an energy representation that is possible for the quantum system (for a quantum system in the singlet state, the time course when the magnetic field is switched on or off has no influence on the state as long as the magnetic field is homogeneous ). However, the switching on and off operation of the ion trap superimposed magnetic field can also be considered as a preparation step. The fact that the temporal course of the magnetic field during the switch-on and switch-off process, for a quantum system in the singlet state, has no influence on the result of these preparation steps, can now be interpreted in such a way that the knowledge about the quantum system is in principle due to these preparation steps can not be changed. The internal properties do not seem to be coupled to the quantum system due to the energetic representation realized at the quantum system, but to be coupled to the quantum system solely on the basis of the knowledge that exists about the quantum system.

The basic idea of the method proposed in this work for the interaction-free entanglement of spatially separated quantum systems in which no classical information channel is required, can now be particularly easily illustrated by the method described in [53] for interaction-free entanglement of quantum systems, if the initial state is the product state Ψ _in = 1/2 (| 0> ₂ + | 1> ₂ ) (| 0> ₁ + | 1> ₁ ). In order to be able to perform the preparation step required for the entanglement in [53], the quantum system must be present as an energetically represented quantum system (the quantum system has an energetic representation possible for the quantum system). For this preparation step to be able to correctly prepare the photons required for the Bell measurement, it must be ensured via the intensity of the laser pulses, which are only about 1 ps long, that both ions can be excited simultaneously and they each emit exactly one photon. Furthermore, the extremely short pulse duration is required to ensure that it is in principle impossible to decide which ion has absorbed which energy from the laser pulse. If one could decide which ion has absorbed which energy, it would in principle be impossible to prepare the photons necessary for the entanglement of the two ions into the required Bell state, because then one would know which photon was emitted by which ion. Since one can not predict in principle which of the quantum system's possible energetic representations has actually been realized, one can only wait for the quantum system to realize one of the desired energetic representations purely by chance. When this is the case, the result of the Bell measurement is clearly displayed. Obtained in the Bell measurement the result is "on both detectors was simultaneously detected per one photon," so it is known that in the quantum system a, the state ratio Ψ _- attributable energy representation realized. This knowledge about the quantum system leads to the fact that now no longer the original product state Ψ _in associated inner properties can be coupled to the quantum system, but now only the inner properties associated with the state Ψ _- can be coupled to the quantum system. The two ions were thus successfully converted into the maximum entangled state -Ψ _- by the Bell measurement.

A quite fundamental difficulty of the experiments described in [531 is that one works there with energetically represented quantum systems and one therefore always has to wait until the quantum system has, by chance, realized one, the energetic representations required for the entanglement. It would be much simpler if one could transform an elementary quantum system into an energetically represented quantum system so that in this preparation step not all possible energetic representations of the elementary quantum system can be realized, but only certain energetic representations can be realized. But how could this be achieved?

Obviously, this would require a selective compulsion to be exercised on the quantum system. According to the considerations in Section V, this could be done by energetically coupling the quantum system to vacuum fluctuations. As considerations of the Casimir effect show, it should be possible to extract any energy from the vacuum for any length of time, provided that the corresponding energy is released back to the vacuum at another point at the same time. A completely analogous process would then also for a quantum system consisting 1/2 systems of two identical spin, conceivable, if one of the quantum system the state share Ψ _- Assign and the quantum system for the state share Ψ _- can realize potential energy representations. The simplest way to clarify this conjecture is by means of an example: If a quantum system consisting of two ⁴⁰ Ca ⁺ ions is prepared, as described in [17], the singlet state 1/2 ^1/2 (| 1> ₂ | 2> ₁ - | 2> ₂ | 1> ₁ ), the quantum system, with the magnetic field switched off, is present as an elementary quantum system. As soon as the magnetic field that can be superimposed on the ion trap is switched on, one ion must absorb the energy ΔE _z / 2 in the spin degree of freedom and the other ion must emit the energy ΔE _z / 2 in the spin degree of freedom. If the switch-on process is now so fast that the switch-on time Δt _s fulfills the condition Δt _s <h / (4πΔE _z / 2) (energy-time-blur relation), then an ion could absorb the energy ΔE _z / 2 by absorbing a created by a vacuum fluctuation, photons record and the other ion the energy .DELTA.E _z / 2, by emitting a photon, to release the vacuum again. Physically, this process could be interpreted as simultaneous flipping of the spins. Therefore, this process could also be called "coherently coupled vacuum fluctuations".

The question then, of course, is whether the process of coherently coupled vacuum fluctuations postulated in Section V can actually occur and, if so, with what probability? As shown in Section VII, this question can be tackled using the spin-statistics theorem. Looking more closely at the spin statistics theorem, there is a fundamental question: what effect can be assigned to the application of the spin statistics theorem to a concrete physical situation and how can this be formulated in the form of a preparation process?

An indication of certain cases is provided by the situation arising on a symmetrical beam splitter when two previously distinguishable, identical quantum systems strike the beam splitter in such a way that they have to be regarded as indistinguishable after the beam splitter. The effect which can be assigned to the application of the spin statistics theorem in this situation can be formally described for bosons by means of the unitary operator U _NLB and for fermions by means of the unitary operator U _NLF (see Eq. Eq. (60)). As shown in Section VII, the preparation processes associated with operators U _NLB and U _NLF must satisfy the following conditions: 1.): In principle, it must be impossible to manipulate a single quantum system and 2.) It must be impossible in principle be able to influence the preparation process as such. It should be noted that the operators U _NLB and U _{NLF transform} an elementary quantum system into an energetically represented quantum system.

A remarkable result of these considerations is that nowhere explicitly states that the individual quantum systems after the beam splitter must be regarded as indistinguishable. This suggests that the operators U _NLB and U _{NLF could} also be realized in situations where the quantum systems remain distinguishable.

If this assumption were true, then it would follow that the process of coherently coupled vacuum fluctuations postulated in Section V must certainly occur in certain situations. The simplest way to clarify these considerations is by means of an example: In the ion trap described in [17], two ⁴⁰ Ca ⁺ ions are prepared as an elementary quantum system in the product state 1/4 ^1/2 (i | 1> ₂ | 1> ₁ + i | 2> ₂ | 2> ₁ + | 2> ₂ | 1> ₁ - | 1> ₂ | 2> ₁ ), this can be converted into an energetically represented quantum system by switching on the magnetic field which can be superimposed on the ion trap. If the on-time Δt _{s is selected to} be short enough to satisfy the condition Δt _s <h / (4πΔE _z / 2), then the process of coherently coupled vacuum fluctuations should occur, following the considerations in Section V. The question is then of course, in which state the quantum system is transferred? If the considerations in Section VII are correct, then the magnetic field switch-on process can be described by a unitary operator of the form U _NLF , since the conditions necessary for the realization of the operator U _NLF are fulfilled. The first condition is fulfilled because the magnetic field was assumed to be homogeneous. Thus, it is in principle impossible to manipulate a single ion by switching on the magnetic field. The second condition is satisfied, since the boundary condition Δt _s <h / (4πΔE _z / 2) ensures that the energetic eigenstates of the ions during the switch-on process can not be assigned a defined energy and therefore no temporal state development can be assigned to the quantum system. As a result, it is impossible in principle to be able to influence the preparation process realized by the switch-on process in a targeted manner. After the switch-on process, the quantum system should therefore be present in the maximally entangled singlet state 1/2 ^1/2 (| 2> ₂ | 1> ₂ - | 1> ₂ | 2> ₁ ) = -Ψ _- as an energetically represented quantum system ,

The process of coherently coupled vacuum fluctuations could thus enable the entanglement of independent spin 1/2 systems without having to resort to an interaction or a classical information channel. This should also be possible if the two spin 1/2 systems are located in different spatially separated ion traps. For this purpose, a binding protocol would have to be created only before commissioning of the superstructures, which determines at which times the magnetic fields superimposed on the respective ion traps are switched on at the same time.

• II. Ein erster Interpretationsansatz: Es gibt wohl keinen anderen Bereich in der Physik, der sich so hartnäckig unserer Intuition entzieht wie die Quantenphysik. Die Quantenphysik ermöglicht uns zwar eine unglaublich präzise Beschreibung der Natur, eine wirklich befriedigende Interpretation liegt allerdings noch immer nicht vor. Die am weitesten verbreitete Interpretation ist die Standardinterpretation [1], deren Begriffe und Vorstellungen ich im Folgenden zugrunde legen möchte. Es werden aber auch andere Ansätze, die sich grundsätzlich von der Standardinterpretation unterscheiden, diskutiert. Beispielsweise die Bohmsche Mechanik [2] oder die Viele-Welten-Interpretation von Everett [1]. Auf diese möchte ich hier aber nicht weiter eingehen, da die grundsätzlichen Fragen im Rahmen der Standardinterpretation klar formuliert werden können. Allein schon die Tatsache, dass bis heute so unterschiedliche Interpretationsansätze diskutiert werden, kann man wohl nur als Indiz dafür auffassen, dass offensichtlich grundsätzliche Zusammenhänge noch nicht befriedigend verstanden sind.

In order to realize a powerful quantum computer, the preparation process used for the interleaving of two quantum systems should be scalable. That is, this should in principle be applicable even if the number of quantum systems involved is arbitrarily large. The process of coherently coupled vacuum fluctuations postulated in Section V could make this possible.

• II. A first interpretation approach: There is probably no other area in physics that stubbornly eludes our intuition like quantum physics. Although quantum physics gives us an incredibly precise description of nature, a truly satisfying interpretation is still not available. The most widely used interpretation is the standard interpretation [1], whose terms and ideas I will base in the following. However, other approaches that differ fundamentally from the standard interpretation are also discussed. For example, the Bohmian mechanics or the Many-Worlds interpretation of Everett [1]. But I do not want to go into that here, since the fundamental questions can be clearly formulated in the framework of the standard interpretation. The very fact that so many different approaches to interpretation are being discussed to this day can only be taken as an indication that obviously fundamental connections are not yet satisfactorily understood.

One reason for this is surely to be seen in the fact that concepts and ideas are shaped by our everyday experience. Naturally, it is not surprising, if one encounters difficulties, to describe phenomena by means of these concepts and representations, which follow laws which deviate from our usual conceptions. But quantum physics is not the first physical theory to put our ideas of the world to the test. For example, if one considers the historical development that led to the formulation of classical electrodynamics [3], one must conclude that it was only with the introduction of the field concept by Faraday that classical electrodynamics in the form of the Maxwell equations became possible. It was only with the introduction of the concept of field that the observed phenomena could be systematically classified and ideas of the underlying laws developed. It is probably difficult to understand how hard it took to introduce the concept of field that is plausible from today's point of view. Another theory, historically closely connected with classical electrodynamics, is the special theory of relativity published by Einstein in 1905 [4]. In contrast to the development of classical electrodynamics, in this case all necessary terms were already available. But even in the case of the special theory of relativity habitual ideas had to be abandoned. For all waves known at that time, experience has shown that they spread in a medium, for example air. It was therefore obvious to postulate that the electromagnetic waves described by the Maxwell equations also propagate in a medium (the ether). However, even then it was known that the Maxwell equations are invariant under transformations that are today called Lorentz transformations, but are not invariant under Galilean transformations. Since classical mechanics is Galileo-invariant, it was suggested that all laws of physics should be invariant under Galilean transformations. The ether hypothesis, however, implied that classical electrodynamics differs from all other areas of physics because there must be an excellent inertial system in which the ether rests. However, all attempts failed to find inertial systems that move relative to the ether. A well-known experiment is the Michelson-Morley experiment [5]. Einstein demanded the invariance of all natural laws under Galileo transformations and thus also the ether hypothesis and postulated in the context of the special theory of relativity that all laws of nature have the same form in all inertial systems and that the speed of light c has the same value in all inertial systems. The requirement that the speed of light has the same value in all inertial systems implies that all laws of nature are Lorentz-invariant. These two postulates are all the more remarkable, considering that in 1905 there was no experimental data confirming these two postulates.

But what is the reason for seeing that quantum physics so stubbornly eludes our intuition, but that we can develop an intuitive understanding of classical electrodynamics and the special theory of relativity, and that we can also resort to a generally accepted interpretation? The main reason for this could be seen in the fact that both in classical electrodynamics and in the special theory of relativity in a concise way, a new, the respective theory auszeichnende idea is clearly recognizable. What is meant by a field, for example, the electric field, is given by an easily understandable, at least idealized, actually feasible measurement regulation. Measure the force on a sample charge at a certain time, at a certain location, and then reduce the size of the sample charge until the measured value for the quotient force per sample charge no longer changes (strictly speaking, the limit is determined when one the size of the sample charge goes to zero). This limit then corresponds to the field strength of the electric field at this time, at this location. For the special theory of relativity, the new, this outstanding idea is formulated in the two postulates clearly understandable. In this sense, classical electrodynamics and the special theory of relativity can be described as complete or completed. For quantum physics, however, it has not yet been possible to specify this "outstanding idea".

The first step in arriving at a satisfactory interpretation for quantum physics is to develop a deeper understanding of the central concepts of quantum physics. The central concept in quantum physics is the concept of state. In the following, only pure states are considered. These can always be described as state vectors in a suitably chosen Hilbert space [1]. Next I would like to use the usual physics Dirac notation [1]. Quite essential properties of quantum physics result from the postulated superposition principle [6], [7]. The superposition principle is understood to mean the following:
If there are several different possibilities (pathways) in the preparation of a state, in such a way that it is not even possible in principle to decide which possibility (pathway) has been realized, then the state resulting from the preparation process is given by, weighted sum of the individual possibilities (ways) weighted with the respective probability amplitude.

For a better understanding, I would like to briefly explain this slightly abstract formulation of the superposition principle with a simple example. The known from the optics lossless symmetrical beam splitter (see ) converts the state Ψ ⁰ = | 1> of the photon arriving from above onto the beam splitter ST1 (path 1 ) in a superposition of ways 2 and 3 to: | 1> → Ψ ¹ = 1 / (2) ^1/2 (| 2> + | 3>) (1)

However, the relation (1) also results quite generally from the requirement that the state transformation caused by a lossless symmetrical beam splitter must be describable by a unitary transformation within the framework of quantum physics [8], [1]. The factor i can be interpreted as π / 2 in the reflection aufretender phase jump with respect to the transmitted portion. The state Ψ ⁰ here stands for the wave train assigned to the individual photon (wave packet). This is determined by the properties of the source used. The wave trains | 1> and | 2> differ from | 3> only in the different propagation directions. The by the two basically indistinguishable possibilities (way 2 or 3 ) are then given by 1 / (2) ^1/2 and i / (2) ^1/2, respectively. As can be seen from this example, the probability amplitudes, in contrast to the positive probability densities exclusively occurring in classical statistical physics, do not have to be positive functions, but can also assume complex values. The two terms must therefore be kept strictly apart. Only by the formation of the sum of sums of the probability amplitudes one obtains statistically interpretable quantities. In this case the probabilities for that on the way 1 Photon striking the symmetric beam splitter after passing through the beam splitter on its way 2 respectively. 3 to detect. The beam splitting property described in (1) applies not only to photons, but more generally to both bosons and fermions.

An immediate consequence of the superposition principle is that quantum systems can interact instantaneously over any distance. However, quantum physics can provide neither a mechanism nor a plausible explanation for this effect. Be the first to have this Facts 1935 Albert Einstein, Boris Podolsky and Nathan Rosen (EPR) in [9]. Quantum systems where these very essential aspects of quantum physics occur are called entangled systems. Cross-linked quantum systems are statistically more correlated than is conventionally possible. Particularly impressive experiments are experiments on teleportation [10] or entanglement swapping [11] with photons. In these experiments, the distances between the individual subsystems can be many kilometers.

In all interlaced quantum systems considered in the literature, a classical information channel is always necessary in order to be able to recognize the entanglement of the subsystems involved. One is confronted with the paradoxical situation that, although mediated by the entanglement of the participating subsystems, these are not connected locally, the knowledge of this entanglement can only be gained through a classical information channel. Looking at the literature, see for example [12], one can not help feeling that this notion is elevated to a postulate. But what is the basis for the idea that a classical information channel is absolutely necessary in order to be able to recognize that two subsystems are parts of an entangled quantum system?

The basic argumentation is the following: In quantum physics, individual measured values are generally unpredictable if several measurement results are possible. Of course, this also applies to entangled systems. Since the statistical distribution of the possible measured values at the other subsystems can not be changed by any kind of measurement on a subsystem, the interleaving of individual subsystems can only be recognized by combining the individual measured values of the participating subsystems and then statistically be evaluated. However, this requires a classic information channel.

Although it is a fact that single measurement results in quantum physics can not even be predicted in principle, if several measurement results are possible, but this fact is a fundamentally unreasonable. For it is not possible in quantum physics to give a description or a succinct idea that makes it possible to explain why a specific measurement result occurs in one specific experiment and not another, if several measurement results are possible. In the context of standard interpretation, this problem then manifests itself as a measurement problem. This is not solved until today. Also, the decoherence theory [1] in which the environment, in particular the measuring device, is included in the quantum physical description can not answer this question. It is also true that by no measurement of any kind on a subsystem, the statistical distribution of the possible measured values on the other subsystems can be changed. For experiments in which only measurements are performed on the individual subsystems, it can therefore be concluded that only then can it be recognized that the subsystems involved are part of an entangled quantum system, if in addition a classical information channel is available. However, it can not be concluded from this that in the context of quantum physics, in principle, a classical channel is required in order to be able to recognize that the subsystems involved are part of an entangled quantum system. For this one would have to be able to show that quantum physics can not provide any other means of transmitting information. Exactly this proof could not be furnished so far however (see also the remarks at the end of section VI).

The very reason for the notion that a classical channel is imperative in order to recognize that two subsystems are parts of an entangled quantum system is rooted in our classical physics experience and the consequent notions. In the context of classical physics, it is imperative to be able to transmit energy from the sender of the information to the receiver of the information. Since, according to the special theory of relativity, the speed of light c has the same value in all inertial systems and matter can not be accelerated to a velocity ≥ c, an exchange of energy is only possible at maximum with the speed of light. Thus, however, an exchange of information can also take place only at maximum speed of light. Although entangled quantum systems can interact instantaneously, this effect is obviously not associated with an exchange of energy between the subsystems involved. If this idea is correct, then information between the participating subsystems can only be exchanged at maximum speed of light by means of a classical channel, since only this one enables the necessary energy exchange.

Now you will not give up this classic idea without good reason. So the question is whether there is experimental evidence in quantum physics that suggests that information can be exchanged between two spatially separated locations without the energy being exchanged between them. As shown by [13] the interaction-free quantum measurement experiments, it is at least possible to obtain information about a location in one of these spatially separate locations without energy being exchanged between the two locations, ie at no time in any form energetically interacted. The principle of the interaction-free quantum measurement can be illustrated particularly clearly with reference to the in schematically illustrated Mach-Zehnder interferometer illustrate. The Mach-Zehnder interferometer consists of two lossless, symmetrical beam splitters and two ideally also lossless mirrors. At the beam splitter ST1, the incident light is first split into two wave trains and superimposed again at the beam splitter ST2 by means of the two mirrors Sp1 and Sp2. If a single photon hits the beam splitter ST1 from above, the state Ψ ^{0 is converted} into a superposition of the two possible paths according to the beam splitter rule (1). If the beam splitter rule (1) is applied to the portions | 2> and | 3> incident on the beam splitter ST2, these are converted into a symmetrical Mach-Zehnder interferometer (both possible paths are exactly the same length) in: | 2> → | 1 (2) ^1/2 (| 4> + i | 5>) (2) | 3> → 1 / (2) ^1/2 (| 5> + i | 4>) (3)

Together with (1), the relationship for a symmetrical Mach-Zehnder interferometer is given by: Ψ ⁰ = | 1> → Ψ ⁴ = i | 5> (4)

That on the way 1 So after striking the Mach-Zehnder interferometer, the photon striking the first beam splitter will certainly be on its way 5 leave again. Thus, once one ignores technical difficulties, the detector D1 will always respond. The detector D2 will therefore never detect a photon in the ideal case. However, this situation changes abruptly when you block one of the two possible paths through an obstacle. Block the way, for example 2 through another detector D3 (see ), it will be on the way 1 photon entering the Mach-Zehnder interferometer is detected by the detector D3 in 50% of the cases, by the detector D2 in 25% of the cases and by the detector D1 in 25% of the cases.

In order to understand why one can speak here of an interaction-free quantum measurement, it is necessary to consider which properties can be attributed to a photon. In the experiments described in [13], the coherence length of the photons used is of the order of 100 μm. The coherence length corresponds approximately to the length of the wave train (wave packet) of a photon along the propagation direction. Perpendicular to the direction of propagation, the area occupied by the individual photons can be determined by the diaphragms, not shown here, used in the generation of the photons. The size of the aperture is in the millimeter range. Thus, the energy of individual photons is safely localized in a space area of about 100 μm in length and about 1 mm in diameter (assuming circular apertures), which travel at the speed of light. Further, it is an experimentally confirmed fact that the energy of a photon is not split at a beam splitter. The energy of a photon is thus either completely reflected at a beam splitter or completely transmitted, whereby it is in principle not possible to predict which case will occur. If photons behave like classical particles, they would turn if the way 2 is blocked by the detector D3, detected in 50% of the cases by the detector D3, in 25% of the cases by the detector D2 and in 25% of the cases by the detector D1. Only when both paths are available in the interferometer does it become apparent that photons are subject to the laws of quantum physics. If photons behave like classical particles, they would be detected by the detector D1 in 50% of the cases and by the detector D2 in 50% of the cases if both paths are possible. However, since photons are subject to the laws of quantum physics, it is certainly known (ideally) that, when a photon is detected by detector D2, an obstacle will guide one of the two possible paths of the interferometer (detector D3 in this example) 2 of the interferometer). If a photon is detected by the detector D2, then you also know with certainty that the energy of the photon on the way 3 has come to the detector D2, since the energy of a photon beam splitters can not be divided. Thus, at no time can the photon interact energetically with detector D3. This is the reason why one speaks in these experiments of an interaction-free quantum measurement. As in the case of classical particles, the photons used will never become energy exchanged between the detector D3 and the detectors D1 and D2. Nevertheless, unlike classical particles, whenever detector D2 responds, one can say for sure that detector D3 is the way 2 blocked.

The experiments described in [13] thus show on the one hand that one can obtain information about a location at a location that is spatially arbitrarily distant from one another, without energy having to be exchanged between these locations. On the other hand, since this information can only be obtained at maximum speed of light, these experiments show that the instantaneous effect possible in quantum physics due to the superposition principle does not necessarily imply an instantaneous transfer of information. Obviously, while it is necessary to abandon the idea that energy must be transmitted from one place to another to transfer information from one place, the idea that this information can be transmitted at maximum speed of light is not called into question by these experiments ,

However, it can also be seen from this example that the fundamental difficulties of understanding in experiments that are carried out with individual quantum systems (in this case individual photons) derive from the concept of state used. The most widely used and also used in the context of the standard interpretation state concept is the following (see, for example, [1], page 30):
"The state of a quantum system is assigned to the special preparation procedure performed. By a quantum state we mean the mathematical (!) Object that allows us to calculate unambiguously the probability of the results of all possible measurements on systems that have undergone the associated preparation procedure. The quantum state thus characterizes the preparation process. So we do not expect the quantum state introduced in this way to have a correspondence in reality that can be assigned to the individual quantum system. "

Strictly speaking, it does not argue that, in principle, it is not possible to assign a state to a single quantum system, but in practice the idea has prevailed that no state can be assigned to a single quantum system. On the basis of this idea, one naturally finds it difficult to develop a deeper understanding of the concept of state for experiments carried out with individual quantum systems, if the interpretation of these experiments is limited exclusively to the statistical level. In order to move from the mathematical object state to a physically interpretable concept of state, which can also be applied to individual quantum systems, it is obvious that a situation that is as simple and as manageable as possible, such as the example of a single photon discussed above, which is a Mach-Zehnder Interferometer crosses to look at. The fundamental question then is what information can be obtained from a measurement of the respective state and how this information is accessible. To address these questions, it is instructive to take a closer look at the superposition principle.

In order to be able to speak of several possibilities within the framework of the superposition principle, it is necessary that the considered quantum system has a distinctive property, by means of which one can recognize the respectively realized possibility. In the example discussed above, this distinctive property of the photon entering the Mach-Zehnder interferometer is evidently given by the energy of the photon. In front of the beam splitter ST1, the energy associated with the photon is uniquely located within the wave train | 1>. The two fundamentally indistinguishable possibilities that arise for the photon incident on the beam splitter ST1 are then represented by the two possible paths that can take the energy of the photon at the beam splitter. One could therefore say that the state Ψ ¹ present after the beam splitter ST1 is "energetically represented" by the two possible paths which can take the energy of the photon. The two possible "energetic representations" are then: (1) the energy of the photon is localized in the wave train | 2> or (2) the energy of the photon is localized in the wave train | 3>. In principle, it can not be predicted which energetic representation was actually realized. The probability with which the individual energetic representations are realized is determined by the magnitude square of the respective probability amplitudes. However, no information about the phases of the individual probability amplitudes is obtained via the energetic representations. This interpretation approach is consistent with the experimentally confirmed facts that a beam splitter can not divide the energy of a photon and the detectors used detect the energy of the photon and convert it into a classically interpretable signal. Therefore, information about the state of the quantum system can only be accessed metrologically via the realized energy representation. Always when one of the detectors D1, D2 or D3 responds (see ), a measurement is made on the quantum system. This simple interpretation approach allows a deeper understanding of the description of the develop interaction-free quantum measurement at the state level. Because: is the way 2 blocked by the detector D3 and then addresses detector D2, so you know for sure that an interaction-free quantum measurement has taken place. In this case, however, the state Ψ ^{1 must} , as soon as the wave train | 2> reaches the detector D3, according to Ψ ¹ = 1 / (2) ^1/2 (| 2> + i | 3>) → Ψ ⁵ = i | 3> (5) be transferred to the state Ψ ⁵ , since no energy and thus no photon was detected at the detector D3. Since the energy of the photon was localized in the wave train | 3> in this case, no measurement is carried out with the arrival of the wave train | 2> at the detector D3, but the state Ψ ^{5 is} prepared. Only in the case that the energy of the photon was located in the wave train | 2> can it lead to a signal at the detector D3. Then, however, the state Ψ ^{1 is} destroyed. Depending on which energetic representation of the state Ψ ^{1 has been} realized in the individual case, the measurement of a quantum system will be either a measurement or a state preparation when the wave train | 2> arrives at the detector D3.

As you can see from this example, this interpretive approach is quite natural. Essential for this is the conceptualization of the "energetic representation". This implies that information about the state of the considered quantum system is only possible via the respectively realized energy representation. All states considered so far are in this sense "energetically represented states". The state Ψ ⁰ (Ψ ⁵ ) is assigned the energetic representation "The energy is located in the wave train | 1> (| 3>)" and the state Ψ ¹ becomes the two possible energetic representations "(1) Wave train | 2> or (2.) The energy is assigned in the wave train | 3> localized ".

In the previous considerations, only the spatial degree of freedom was considered. With this interpretation approach, however, experiments can be described in which the spin of the photons must be considered. For example, the above-mentioned experiment for teleportation [10]. In the experiment described there, the involved photons are entangled over the spin-degree of freedom (the polarization). In order to be able to recognize that the process of teleportation was successful, a completely polarizing beam splitter is used on the receiver side ("BOB"). This entangles the spin degree of freedom with the spatial degree of freedom of the incident photon on this. By means of the fully polarizing beam splitter thus information on the degree of spin freedom on the spatial degree of freedom are accessible. Again, the information about the state of the quantum system is represented by which path the energy of the photon takes at the beam splitter. Whether the energy of the photon is reflected or transmitted at the beam splitter. But even for interference experiments with quantum systems that have a non-zero rest mass, such as molecules or neutrons, energetic representations can be assigned to the corresponding states. For this, it is only necessary to replace the energy associated with the observed photons by the kinetic energy associated with the mass and velocity of the particles under consideration. Corresponding interference experiments with molecules or neutrons are described, for example, in [14] and in [15]. Even for particles which have a spin, such as neutrons, it is possible to entangle the degree of freedom of spin with the spatial degree of freedom and thus to obtain information about the degree of spin-freedom by means of the spatial degree of freedom. I'll talk about that later.

• III. Eine energetische Betrachtung einfacher Modellsysteme: Die Energie die in einem klassischen System beispielsweise als kinetische oder potentielle Energie gespeichert ist, kann im Prinzip zu jedem Zeitpunkt genau angegeben werden. Was kann man aber über die Energie eines einzelnen Quantensystems das in einem definierten Zustand präpariert wurde aussagen? Als Untersuchungsobjekt bieten sich hier einzelne Atome oder Ionen an. Experimente an einzelnen Atomen oder Ionen bieten die Möglichkeit gerade grundsätzliche Fragen zur Interpretation der Quantenphysik [16] an einem gut überschaubaren einfachen Quantensystem zu diskutieren. Experimente mit einzelnen Ionen [17] sind heutzutage mit hoher Präzision möglich. Die technische Grundlage für Experimente mit einzelnen Ionen wurde durch die Entwicklung geeigneter Ionenfallen gelegt [18]. Erst hierdurch wurde es möglich einzelne Ionen über einen längeren Zeitraum zu speichern, gezielt in definierten Zuständen zu präparieren und die erzeugten Zustände zu analysieren. Um einen direkten Bezug zu den in [17] beschriebenen Experimenten zu ermöglichen, möchte ich die dort eingeführten Bezeichnungen und Symbole hier übernehmen und die Ionenfalle kurz beschreiben.

If one assumes that the interpretation approach proposed above is generally valid and this assumption is corroborated by the examples considered below, a quite fundamental question arises. Obviously, the state of a quantum system is neither completely nor unambiguously described by the energetic representations assigned to it. As I will show below, it is also possible to prepare quantum systems, the state of which can only be interpreted as meaning that it has no energetic representation, but can at any time be transformed into an energetically represented state. Whereby the reverse preparation step is possible at any time. However, if it is possible to prepare quantum systems in such a way that the state has no energetic representation, then of course the question arises as to what role the energetic representations play in state preparation and how the properties of a quantum system described by the state are to be interpreted , The clarification of these questions seems to me to be the key to a satisfactory interpretation and the formulation of an idea fundamentally characterizing quantum physics.

• III. An energetic consideration of simple model systems: The energy that is stored in a classical system, for example, as kinetic or potential energy, can in principle be specified exactly at any time. But what can one say about the energy of a single quantum system that has been prepared in a defined state? Here, single atoms or ions may be used as the examination object. Experiments on single atoms or ions offer the possibility to discuss basic questions concerning the interpretation of quantum physics [16] in a well-ordered simple quantum system. Experiments with single ions [17] are possible today with high precision. The technical basis for experiments with single ions was laid by the development of suitable ion traps [18]. Only then was it possible to store individual ions over a longer period of time, to purposely prepare them in defined states and to analyze the generated states. In order to make a direct reference to the experiments described in [17] possible, I would like to adopt the designations and symbols introduced there and briefly describe the ion trap.

The linear ion trap used in [17] offers the possibility to store single but also several ions. The experiments use ⁴⁰ Ca ⁺ ions. The structure of the ion trap is shown schematically in FIG played. By means of four blade-shaped electrodes, to which a high-frequency alternating electrical voltage is applied, the possibility of movement of the ions stored in the trap is limited to one degree of freedom - the movement along the x-axis. In order to hold the ions in the trap, a further positively charged electrode is arranged at both ends of the trap. The parameters of the ion trap are selected so that the localization of the ions stored in the trap is approximately 1 μm and the distance between the ions is approximately 5 μm. The ion trap is superimposed in the z-direction still a constant homogeneous magnetic field. This splits the energy levels of the ⁴⁰ Ca ⁺ ions used via the anomalous Zeeman effect and determines the quantization direction. The ion trap is operated in a vacuum at about 10 ^-9 Pa in order to exclude collisions between the ions and the residual gas molecules.

The simplest nontrivial quantum systems can be realized by systems with two energy levels. These are often referred to as quantum bits short qubits. shows a simplified energy level scheme of the ⁴⁰ Ca ⁺ ions used. The relevant energy levels for the discussion are the ground state | 1>, the long-lived first excited state | 0> and the short-lived excited state | P>. The qubit is realized by means of the two states | 1> and | 0>. These form an ONB (orthonormal basis) in a two-dimensional Hilbert space H ₂ . Any pure state Ψ of a qubit can then be described as an element of the Hilbert space H ₂ by Ψ = a ₀ | 0> + a ₁ | 1>, with | a ₀ | ² + | a ₁ | ² = 1 (6) where a ₀ and a _{1 are} complex numbers. If both coefficients a ₀ and a _{1 are} not equal to zero, then one speaks of a superposition state. The state | 0> can be excited with the manipulation laser at a wavelength of 729 nm starting from the ground state | 1>. The state | P> is used for state analysis. It can also be excited with the analysis laser at a wavelength of 397 nm starting from the ground state | 1>. The two possible measurement results in the state analysis are then "ion-fluoresces" (the transition from | 1> to | P> can be excited) and "ion does not fluoresce" (the transition from | 1> to | P> can not be excited) , If the ion is in the ground state | 1> before the state analysis, the transition from | 1> to | P> can be excited and the result obtained is "ion fluoresced". If the transition | 1> to | P> can not be excited because the ion was in the state | 0>, for example, before the measurement, the result obtained is "Ion does not fluoresce".

The prerequisite for being able to prepare several ions in an entangled state is to be able to prepare a single ion defined in a desired state. The preparation method used in [17] can be regarded as analogous to the preparation method for spin systems known from NMR [22] (nuclear magnetic resonance). The equations of motion of a spin ½ system were first discussed in detail by F. Bloch [32] and are therefore also called Bloch equations. Feynman, Vernon, and Helwarth [19] were able to show that the equations of motion of each two-level system are formally consistent with the equations of motion of a spin ½ system. This formal correspondence also makes it possible to apply the descriptive image of the Bloch sphere to the two-level optical systems using the Bloch vector to describe the states of a spin ½ system. Each state can thus be represented graphically as a point on the surface of the Bloch sphere (the top of the Bloch vector) [1], [19]. For example, is an ion starting from the ground state | 1> by means of a π / 2 pulse (R _{1, -y} (π / 2)) in the superposition state Ψ ⁷ = 1/2 ^1/2 (| 0> + | 1>) (7) prepared, this corresponds to the rotation of the Bloch vector by 90 ° about the -e _y -axis of the Blochkugel. The Bloch vector, which initially showed in the -e _z direction, then points in the e _x direction. Because of this analogy, two-level systems are also commonly referred to as pseudo-spin 1/2 systems, and in the case of 2-level optical systems, the block equations are called optical block equations. For these, the probability of finding an ion after a state measurement in the ground state | 1> or in the excited state | 0> is determined via the z-component of the Bloch vector. The x and y component then describes the quantum physical coherence between these two states and can be physically interpreted as an oscillating electrical moment. About the phase of the laser pulses relative to the oscillating moment of the ions, the axis of rotation of the Bloch vector can be determined. The rotation angle of the Bloch vector can be specified via the pulse duration and the pulse intensity [17]. By means of state tomography [17] the respectively prepared state of the system can be analyzed.

• I): Wenn man zu Beginn das Ion im Grundzustand |1> präpariert und dann das Ion mit dem Analysenlaser beleuchtet, weiß man mit Sicherheit, dass man als Messergebnis „Ion fluoresziert” erhält. Wird das Ion zu Beginn im Grundzustand präpariert und danach mit dem Manipulationslaser mittels eines π-Pulses (R_1,–y(π)) beleuchtet, dann weiß man ebenfalls mit Sicherheit (wenn man einmal von experimentellen Unsicherheiten absieht), dass man als Messergebnis „Ion fluoresziert nicht” erhält, da das Ion nach dem π-Puls im angeregten Zustand |0> vorliegt.
• II): Präpariert man das Ion in einem Überlagerungszustand, so kann ein einzelnes Messergebnis prinzipiell nicht mehr vorhergesagt werden. Man kann für ein einzelnes Messergebnis nur noch eine über das Betragsquadrat der Amplituden a₀ und a₁ festgelegte Wahrscheinlichkeit angeben. Präpariert man beispielsweise das Ion ausgehend vom Grundzustand |1> mit dem Manipulationslaser mittels eines π/2-Pulses (R_1,–y(π/2)) in den Überlagerungszustand Ψ⁷, so treten die beiden Messergebnisse „Ion fluoresziert” und „Ion fluoresziert nicht” in diesem Fall mit gleicher Wahrscheinlichkeit ein. D. h., in 50% der Fälle erhält man das Ergebnis „Ion fluoresziert” und in 50% der Fälle das Ergebnis „Ion fluoresziert nicht”.

For a single ion in the ion trap, the following experimentally confirmed facts are obtained:

• I): If one prepares the ion in the ground state | 1> at the beginning and then illuminates the ion with the analysis laser, one knows with certainty that one receives "ion fluorescence" as measurement result. If the ion is initially prepared in the ground state and then illuminated with the manipulation laser by means of a π-pulse (R _{1, -y} (π)), then one knows with certainty (apart from experimental uncertainties) that one as a measurement result "Ion does not fluoresce" because the ion is present after the π-pulse in the excited state | 0>.
• II): If the ion is prepared in a superposition state, a single measurement result can in principle no longer be predicted. For a single measurement result, it is only possible to specify a probability determined by means of the absolute value squares of the amplitudes a ₀ and a ₁ . If, for example, the ion is prepared from the ground state | 1> with the manipulation laser by means of a π / 2 pulse (R _{1, -y} (π / 2)) into the superposition state Ψ ⁷ , the two measurement results "ion fluoresces" and " Ion does not "fluoresce" in this case with equal probability. In other words, in 50% of the cases the result "ion fluoresces" and in 50% of the cases the result "ion does not fluoresce".

Now the question arises, how the two possible measurement results "Ion fluoresces" and "Ion does not fluoresce" can be interpreted?

If one has obtained "Ion fluorescence" as measurement result, one knows with certainty that the ion is in the ground state | 1> after the measurement. If the measurement result is "Ion does not fluoresce", then one knows with certainty that the ion is present in the excited state | 0> after the measurement. However, in order for an ion to be in the excited state after the measurement result "Ion does not fluoresce", the ion must at some point have a quantum of light (with the energy hf ₀₁ , where h is Plank's constant and f ₀₁ is the transition of | 1> after | 0> resonant excitation transition frequency are called) have absorbed. Therefore, if one prepares the ion in an arbitrary state from a ground state | 1> by a further preparation step and obtains the result "ion does not fluoresce", the ion must have absorbed the energy of a light quantum (hf ₀₁ ) at some point during this preparation step yes, only the manipulation laser can provide light quanta with the appropriate energy (hf ₀₁ ). Here it was assumed that a laser pulse can interact with an ion only as long as the laser pulse spatially overlaps with the ion and only during this interaction time (the pulse duration) energy between the laser pulse and the ion quantized in the form of light quanta can be exchanged.

Thus, the following interpretation (IN1) can be given for the two measurement results "Ion fluoresces" and "Ion does not fluoresce":
IN1): If one obtains the measurement result "Ion does not fluoresce", one knows with certainty that the ion has the energy hf ₀₁ stored (a light quantum with the energy hf ₀₁ taken). If one obtains the measurement result "Ion fluoresces", then one knows with certainty that the ion has stored no energy. In other words, the two possible measurement results unambiguously represent the information regardless of the prepared state of the ions whether a specific ion has stored energy ("ion does not fluoresce") or whether a specific ion has not stored any energy ("ion fluoresces").

The energy assigned to the ground state | 1> has been set equal to zero here. The result is an informal interpretation (IN2) for the states | 1> and | 0>:
IN2): If an ion is prepared in the ground state | 1>, the state | 1> represents the information "Ion has no energy stored". On the other hand, if one prepares an ion in the excited state | 0>, the state | 0> then represents the information "Ion has stored the energy hf ₀₁ .

But how is a superposition state to be interpreted?

According to II.) It is a fact that for an ion that was prepared in a superposition state, individual measurement results can not be predicted in principle. So you can not have any information about whether a specific ion has the energy hf ₀₁ or not. Because if you had at least in principle the ability to decide whether or not a single ion has stored the energy hf ₀₁ , you could also predict every single measurement result. But what is not possible according to II.).

This results in the following interpretation (IN3) for superposition states:
IN3): A superposition state represents the information that there are two basically indistinguishable possibilities: 1.): "Ion has stored the energy hf ₀₁ " or 2.): "Ion has no energy stored". The probability of the two possibilities is given by the magnitude square of the amplitudes a ₀ and a _{1 of} the two possibilities.

If, for example, an ion is prepared in the superposition state Ψ ⁷ starting from the ground state | 1> by means of a π / 2 pulse (R _{1, -y} (π / 2)), then the prepared state is clearly established. But the question "Ion has stored the energy hf ₀₁ or has no energy stored" can not be answered in principle. You can only specify a probability of 50% for this. Only by a state measurement can be decided whether a specific ion has stored energy or not. However, the state of superposition Ψ ⁷ is inevitably transferred to the state | 1> or | 0> by the state measurement, depending on which measurement result has been obtained.

This lays the foundation for the interpretation of entangled states.

The description of composite qubit systems allows the product Hilbert space, which is defined by the tensor product of the Hilbert dreams of the individual qubits. For a composite system of two qubits, the four-dimensional product Hilbert space H ₄ = H ² ₂ × H ¹ ₂ results, where H ⁱ ₂ , i = 1,2 denote the two two-dimensional Hilbert spaces of the individual subsystems. A state is referred to as entangled in this product Hilbert space, if there are no elements φ ₁ εH ¹ ₂ and φ ₂ εH ² ₂ , so that Ψ in the form Ψ = φ ₂ φ ₁ (8) ie can be written as a product [1]. For the discussion of entangled states, two ions in the ion trap are considered below. This does not limit the generality of the statements made. All conclusions are also valid for entangled systems with any number of ions.

A well-known example of an entangled state is the Bell state Ψ _- ²¹ = 1/2 ^1/2 (| 1.0> - | 0.1>) = 1/2 ^1/2 (| 1> ₂ | 0> ₁ - | 0> ₂ | 1> ₁ ) ( 9)

This maximum entangled state is also discussed in [17] and the preparation procedure is explicitly stated. In order to be able to describe the preparation steps, it is necessary to introduce a further quantum number v, which describes the common fundamental vibration of the ions located in the ion trap. This fundamental vibration, in which all ions oscillate in common mode ("the distance between the ions remains constant") at the frequency f ₊ along the x-axis of the ion trap, can be excited by passing any ion by means of the manipulating laser at the frequency ( f ₀₁ + f ₊ ). The frequency (f ₀₁ + f ₊ ) is also called the blue sideband of the carrier frequency f ₀₁ which resonantly excites the transition from | 1> to | 0>. v = 0 then means that the common fundamental is not excited. v = 1 means that the common fundamental is excited. The ground state then results in Ψ ₀ = | 1.1, v = 0>. Both ions are in the ground state and the common fundamental is not excited. The left ion in the ion trap is referred to as ion 2 and the right as ion 1. If, for example, starting from the ground state Ψ ₀ = | 1,1, v = 0>, the left ion is illuminated with a π-pulse on the blue sideband (R ⁺ _{2, -y} (π)), then the ground state Ψ ₀ = | 1,1, v = 0> transferred to the state | 0,1, v = 1>. Thus, the ion 2 is simultaneously prepared in the excited state | 0> and the common fundamental excited. If, starting from the ground state Ψ ₀ = | 1.1, v = 0>, the right ion is illuminated with a π-pulse on the carrier frequency (R _{1, -y} (π)), the ground state Ψ ₀ = | 1.1v = 0> transferred to the state | 1,0, v = 0>. Now only the ion 1 is prepared in the excited state | 0>.

• 1.) Schritt: Das rechte Ion wird mit einem resonanten π-Puls (R_1,–y(π)) auf der Trägerfrequenz beleuchtet. Dadurch wird der Grundzustand Ψ₀ = |1,1v=0> in den Zustand |1,0,v=0> überführt.
• 2.) Schritt: Das linke Ion wird mit einem π/2-Puls (R⁺ _2,–y(π/2)) auf dem blauen Seitenband beleuchtet. Dadurch wird der Zustand |1,0,v=0> in den Zustand 1/2^1/2(|1,0,v=0> + |0,0,v=1>) überführt.
• 3.) Schritt: Das rechte Ion wird mit einem π-Puls (R⁺ _1,–y(π)) auf dem blauen Seitenband beleuchtet. Dadurch wird der Zustand 1/2^1/2(|1,0,v=0> + |0,0,v=1>) in den Zustand 1/2^1/2(|1,0,v=0> – |0,1,v=0>) überführt. Dieses ist der gewünschte Bellzustand Ψ_– ²¹.

The maximum entangled Bell state Ψ _- ²¹ can be prepared in three steps starting from the ground state Ψ ₀ = | 1.1, v = 0> [17]:

• 1.) Step: The right ion is illuminated with a resonant π-pulse (R _{1, -y} (π)) at the carrier frequency. This converts the ground state Ψ ₀ = | 1,1v = 0> into the state | 1,0, v = 0>.
• 2.) Step: The left ion is illuminated with a π / 2 pulse (R ⁺ _{2, -y} (π / 2)) on the blue sideband. This converts the state | 1,0, v = 0> to the state 1/2 ^1/2 (| 1,0, v = 0> + | 0,0, v = 1>).
• 3.) Step: The right ion is illuminated with a π-pulse (R ⁺ _{1, -y} (π)) on the blue sideband. This will turn 1/2 ^1/2 (| 1.0, v = 0> + | 0.0, v = 1>) into 1/2 ^1/2 (| 1.0, v = 0> - | 0,1, v = 0>). This is the desired bell state Ψ _- ²¹ .

How can the bell state Ψ _- ^{21 be} interpreted? It is instructive to take a closer look at the individual preparation steps.

The initial state for the preparation procedure of the Bell state Ψ _- ²¹ is the ground state Ψ ₀ = | 1,1, v = 0>. In the ground state no energy is stored in the system, since both ions are in the ground state and the common fundamental vibration of the ions is not excited. In the first preparation step, the right ion is illuminated with a resonant π-pulse (R _{1, -y)} (π)) at the carrier frequency. This prepares the right ion in the excited state | 0>. So you know with certainty that the right ion has stored the energy hf ₀₁ . In the second preparation step, the left ion is illuminated with a π / 2 pulse (R ⁺ _{2, -y} (π / 2)) on the blue sideband. The left ion is thereby prepared in a superposition state. This represents (IN3) the information that there are two indistinguishable possibilities: 1.) "Ion has stored the energy hf ₀₁ " or 2.) "Ion has no energy stored". However, since the system can only record energy quanta of the energy h (f ₀₁ + f ₊ ) from the π / 2 pulse (R ⁺ _{2, -y} (π / 2)) on the blue sideband, the left ion can only then Stored energy, even if the common fundamental vibration was stimulated. Thus, for the system the statement is: There are two indistinguishable possibilities: 1.) "The left ion has the energy hf ₀₁ stored and the energy stored in the common fundamental is hf ₊ " or 2.) "The left ion has none Energy stored and it was the common fundamental vibration not excited. In the third preparation step, the right ion is illuminated with a π-pulse (R ⁺ _{1, -y} (π)) on the blue sideband. This can de-excite the right ion into the ground state if and only if the common fundamental is excited. If no energy was stored in the system in the second step, the third step has no effect, because then the right ion can not be converted to the ground state. If, on the other hand, the energy h (f ₀₁ + f ₊ ) was stored in the system in the second step, the common fundamental is excited and the right ion can be converted into the ground state together with the common fundamental. In this case, the system delivers the energy h (f ₀₁ + f ₊ ) to the laser pulse in the form of an energy quantum.

The following interpretation (IN4) results for the Bell state Ψ _- ²¹ :
IN4): The Bell state Ψ _- ²¹ represents the information that there are two basically indistinguishable possibilities: 1.): "Ion 1 has stored the energy hf ₀₁ and Ion 2 has no energy stored" or 2.): "Ion 1 has no energy stored and ion 2 has stored the power hf ₀₁ ".

For the Bell state Ψ _- ^21, one knows with certainty that the system has stored the energy hf ₀₁ . But the question in which ion the energy hf _{01 is} stored can not be answered in principle. You can only give one probability for this. In 50% of the cases the energy hf ₀₁ is stored in the ion 1 and in 50% of the cases the energy hf _{01 is} stored in ion 2. It is immediately clear that in this way any arbitrary entangled state can be interpreted. For the Bell condition Ψ _- ²¹ = 1/2 ^1/2 (| 0,0> - | 1,1>) (10) For example, the result is the interpretation: The state φ. ²¹ represents the information that there are two fundamentally indistinguishable possibilities: 1.): "Ion 1 has stored the energy hf ₀₁ and ion 2 has stored the energy hf ₀₁ " or 2.): "Ion 1 has no energy stored and ion 2 has no energy stored ". Both possibilities exist with the same probability of 50%.

One can thus prepare any entangled state between two or more ions. The essential step for the preparation of an entangled state is always to prepare individual ions in a suitable superposition state by means of an appropriately selected pulse on the blue sideband. However, since the system can only record energy quanta of the energy h (f ₀₁ + f ₊ ) from the laser pulse on the blue sideband, the respective ion can only have stored energy if and only if the common fundamental has been excited. Thus, ignorance of whether the ion prepared in the superposition state has stored energy or not is also extended to the degree of freedom of the system's common vibration along the x-axis. Illuminating now For example, any other ion that was previously known to have stored the energy hf ₀₁ with a π-pulse (R ⁺ _{1, -y} (π)) on the blue sideband (in the preparation process described above). State Ψ _- if this was the 3rd step), then with this preparation step one also loses information about the energy stored in this ion. In this way, the information about the stored energy in the individual ions is lost step by step and you get in the end the desired entangled state.

As the examples explicitly explained here for the preparation process of the states Ψ ⁷ and Ψ _- ²¹ show, one can generally apply the interpretation approach suggested above to these model systems as well. The required characteristic of these quantum systems is given by the energy stored in the individual ions. The two possible energetic representations can therefore be assigned to state Ψ ^7: 1. "The ion has stored the energy hf ₀₁ " or 2.) "The ion has no energy stored" and the state Ψ _- ²¹ can be used for the two possible energetic representations 1. ) "Ion 1 has stored the power hf ₀₁ and ion 2 has no power stored" or 2.): "Ion 1 has no power stored and ion 2 has been assigned the power hf ₀₁ stored". The states | 0> and | 1> are then assigned the energetic representation "The ion has stored the energy hf ₀₁ " or "The ion has stored no energy". The probability with which the respective energetic representations are present is determined by the absolute value squares of the probability amplitudes assigned to them.

At this point I would like to briefly comment on the measuring problem already mentioned above. For all previously considered quantum systems, the measurement problem can be understood as an immediate, logical consequence of the superposition principle. Because: If several measurement results are possible during a measurement, then the quantum system under consideration is assigned several energy representations by means of the superposition principle. The information obtainable in a measurement on a quantum system is then given by the actually realized energetic representation. As a result of a measurement, one always only receives information about which energetic representation has actually been realized. Since it is not possible in principle to predict which energetic representation has been realized due to the superposition principle, individual measurement results can not be predicted.

This fact can also be understood as follows: For example, in order to be able to prepare an ion in a defined superposition state, the preparation process must be designed in such a way that it is fundamentally impossible to predict whether a specific ion has exchanged energy with the laser beam of the manipulation laser or not , In the experiments described here, the ions are illuminated at an intensity of about 500 W / cm ² . The intensity is so high that it is not possible in principle to decide whether the ion has exchanged energy with the laser beam of the manipulation laser or not. But it is also not possible to predict whether a single ion has stored the energy hf ₀₁ or not. Since the possible measurement results "Ion does not fluoresce" or "Ion fluoresces" but are determined by whether the ion has stored the energy hf ₀₁ or has stored no energy, therefore, individual measurement results can not be predicted in principle. The deeper reason for the fact that individual measurement results can not even be predicted in principle could thus also be seen in the fact that the energy stored in the individual ions is quantized. Only then does it become possible under the superposition principle to unambiguously assign the unique property of the quantum system required for its application to the energy stored in the respective ion.

But how can the properties of a quantum system described by the mathematical object state be physically interpreted? Each state can be considered as an element of a vector space. For example, the state Ψ _- ^{21 is} then an element of the vector space H ₄ = H ² ₂ × H ¹ ₂ . The base (B1) used above is by the basis vectors | 0,0>, | 0,1>, | 1,0>, | 1,1> (B1) given. Now you can choose in the vector space H ₄ but also a different basis. For example, the basis (B2) given by the basis vectors, Φ ₊ ²¹ = 1/2 ^1/2 (| 0,0> + | 1,1>) Ψ ₊ ²¹ = 1/2 ^1/2 (| 1,0> + | 0,1>) Φ _- ²¹ = 1/2 ^1/2 (| 0,0> - | 1,1>) Ψ _- ²¹ = 1/2 ^1/2 (| 1,0> - | 0,1>) (B2) represented in the base (B1). However, the basis (B1) is characterized by the fact that this is the only physically directly accessible basis. Because: The information accessible during a measurement is given by the energy stored in the individual ions. However, as soon as one knows whether the individual ions have stored energy or have stored no energy, they must be present in the respective energetic eigenstates | 0> or | 1> after the measurement. This fact is also called projection postulate [1]. One then speaks of a collapse of the quantum physical state. For this quantum system consisting of two ions, however, all possible measurement results are described by the four basis vectors of the base (B1). It is therefore fundamentally not possible, for example, to obtain the information as the result of a measurement "The quantum system is in the state Ψ _- ²¹ ". For this, the quantum system would have to be in the state Ψ _- ²¹ after the measurement. Which is obviously not possible. As in the literature [1], I would like to refer to basic systems that are directly accessible in terms of metrology in the following as calculation basis.

und

und der Operator U_B2B1 ist dann durch

gegeben. Wie man leicht nachprüfen kann, werden dann die einzelnen Basisvektoren der Basis (B2) durch den Operator U_B2B1 nach Φ₊ ²¹ → 10,0> Φ_– ²¹→ |0,1> Ψ₊ ²¹ → |1,0> Ψ_– ²¹ → |1,1> (15) auf die Basis (B1) abgebildet. Somit wird beispielsweise der Zustand Ψ₊ ²¹ mittels des unitären Operators U_B2B1 eineindeutig in den Zustand |1,0> überführt. Wird danach an dem Quantensystem eine Messung durchgeführt, so weiß man mit Sicherheit, dass man als Messergebnis die Information erhält „Das rechte Ion hat die Energie hf₀₁ gespeichert und das linke Ion hat keine Energie gespeichert”. Allerdings verliert man hierbei jegliche Information über die tatsächlich realisierte energetische Repräsentation des Quantensystems im Zustand Ψ₊ ²¹. In diesem Sinne kann man jedem der vier Bell-Zustände der Basis (B2) eineindeutig einen zu diesem „formal äquivalenten Zustand” der Rechenbasis (B1) zuordnen. Somit kann man aber auch jedem beliebigen Zustand Ψ einen bezüglich des Operators U_B2B1 zu diesem formal äquivalenten Zustand Ψ' zuordnen. Der zu einem Zustand Ψ bezüglich des Operators U_B2B1, der die Basis (B2) eineindeutig auf die Rechenbasis (B1) abbildet, formal äquivalente Zustand Ψ' ergibt sich dann durch Anwendung des Operators U_B2B1 auf den Zustand Ψ. Das einem formal äquivalenten Zustand zugeordnete Quantensystem möchte ich im Folgenden als „formal äquivalentes Quantensystem” bezeichnen.But if in H4 the only metrologically directly accessible basis is given by the basis (B1), which physical meaning can then be assigned to the other possible basis systems, for example the basis (B2)? Now every base of a vector space can be transformed into a different one of the bases of this vector space by a one-to-one mapping. If it is possible to describe a basal change by a unitary operator (Figure) and to succeed in physically implementing it by means of suitable preparation steps, then one can also assign a physical meaning to the new base thus accessible. For the base (B2) this is possible. The basis (B2) can be based on (B1) by means of the unitary operator U _B2B1 = H ₁ U ¹² _CNOT (11) be imaged. Here, U ¹² _{CNOT denotes} the CNOT gate acting on both ions (Ion 1 and Ion 2) and H ₁ the Hadamard operator acting only on Ion 1 [20]. The operators U ¹² _CNOT and H ₁ are then given by

and

and the operator U _B2B1 is then through

given. As can easily be verified, the individual basis vectors of the base (B2) then become the operator U _B2B1 Φ ₊ ²¹ → 10.0> Φ _- ²¹ → | 0.1> Ψ ₊ ²¹ → | 1.0> Ψ _- ²¹ → | 1.1> (15) on the basis (B1). Thus, for example, the state Ψ ₊ ^{21 is} _{unambiguously converted} to the state | 1.0> by means of the unitary operator U _B2B1 . If a measurement is then carried out on the quantum system, one knows with certainty that the result obtained is the information "The right ion has stored the energy hf ₀₁ and the left ion has no energy stored". However, one loses any information about the actually realized energetic representation of the quantum system in the state Ψ ₊ ²¹ . In this sense, one of the four Bell states of the base (B2) can be uniquely assigned one to this "formally equivalent state" of the calculation base (B1). Thus, however, one can associate any state Ψ with respect to the operator U _B2B1 to this formally equivalent state Ψ '. The formally equivalent state Ψ 'for a state Ψ with respect to the operator U _B2B1 , which _{uniquely represents} the base (B2) on the _computing base (B1), then results from the application of the operator U _B2B1 to the state Ψ. In the following, I would like to call the quantum system assigned to a formally equivalent state a "formally equivalent quantum system".

Using the quantum systems formally equivalent to a quantum system, it is thus possible to obtain information about the state of the quantum system. This is used, for example, in the state tomography (state analysis) already mentioned above in [17]. The state tomography is particularly clear if one considers only a single ion in the ion trap, since then the state can be described by the Bloch vector (see above). For the state analysis described in [17] the required formally equivalent quantum systems are realized by rotations of the Bloch vector by 90 °, with individual normal vectors of the Bloch sphere (-e _x and e _y ) as axes of rotation. As part of the state analysis, the quantum system under consideration, of which the state was unknown prior to the state analysis, can be assigned a state step by step, starting from the energetic representations of the quantum system that are directly accessible via the computing base, then using the formally equivalent quantum systems. Herein also the physical meaning of the different basic systems is to be seen. A state analysis of the basis (B2) formed by the Bell states is referred to in the literature [1] as the Bell state analysis.

However, the question arises whether the properties assigned to a quantum system via the formally equivalent states can always be understood as physically realized properties of this quantum system. What I mean by this is that in order to be able to say that a physical property can be attributed to a certain physical property, it must in some way also be accessible by measurement. For the quantum systems considered here, the only metrologically accessible information about the respectively realized energetic representations is given. However, only then can a quantum system be assigned a specific property in the sense of a physical property if the quantum system can be assigned the corresponding energetic representations with regard to the superposition principle. In this sense, one can then speak of the fact that a property assigned to the state can also be assigned to the quantum system as a physically realized property.

If a quantum system is prepared, for example, in the Bell state Ψ ₊ ²¹ , then this can be converted by means of the operator U _B2B1 into the state | 1.0> which is formally equivalent with respect to the operator U _B2B1 . If one then carries out a measurement on the formal equivalent quantum system prepared in this way, the result obtained is certainly "The right ion has stored the energy hf ₀₁ and the left ion has stored no energy". However, this information now allows the quantum system, except for an unobservable phase, to assign the property "The quantum system is in state Ψ ₊ ²¹ ". This property can then be understood as a physically realized property, since the energetic representations possible for the quantum system in the sense of the superposition principle correspond to this property. The same applies to the other Bell states of the base (B2). In general, however, it is not possible to _assign a quantum system, which is accessible via the _properties that are formally equivalent with respect to the operator U _B2B1 , as physically implemented properties.

I would like to clarify this situation with a simple example. If one sets up a quantum system in the state | 1,1>, then the only possible energetic representation is given by "both ions have no energy stored". The state | 1,1> can be written with respect to the base (B2) as | 1.1> = 1 / (2) ^1/2 (Φ ₊ ²¹ - Φ _- ²¹ ). (16)

The conditionally equivalent to | 1,1> with respect to the operator U _B2B1 is then given by Ψ ₁₇ = U _B2B1 (| 1,1>) = 1 / (2) ^1/2 (| 0,0> - | 0,1>) (17)

If a measurement is carried out on the formally equivalent quantum system in state Ψ ₁₇ , then in 50% of the cases the measurement result "ion 1 and ion 2 fluoresce" and in 50% of the cases the measurement result "ion 1 does not fluoresce and ion 2 fluoresces. " Although the property can thus be assigned to the quantum system "The state can be written as a linear combination of the basis vectors Φ ₊ ²¹ and Φ _- ²¹ and the magnitude square of the respective probability amplitudes is equal to 1/2". However, one can do this, in Eq. (16) immediately recognizable property of the quantum system in the state | 1,1> does not regard it as a physically realized property in the sense of the superposition principle. For this, the quantum system would have to have the energetic representations associated with the basis vectors Φ ₊ ²¹ and Φ _- ²¹ . Which is not the case. In the following, I would like to call the properties of a quantum system that are not physically realized in this sense "inherent properties" of a quantum system.

The state description of a quantum system in the state | 1,1> with respect to the Bell basis must therefore be regarded as an inherent property of the quantum system. That of a quantum system in state Ψ ₊ ^21, however, as a physically realized property. However, one can not conclude from this that only the maximum entangled Bell states have no inherent properties regarding the description of the state in the Bell basis. I would like to clarify this with an example:
The product state Ψ ₁₈ = 1/2 ^1/2 (| 1> ₂ + | 0> ₂ ) 1/2 ^1/2 (| 1> ₁ + | 0> ₁ ) can be written with respect to the base (B2) as Ψ ₁₈ = 1/2 ^1/2 (| 1> ₂ + | 0> ₂ ) 1/2 ^1/2 (| 1> ₁ + | 0> ₁ ) = 1/4 ^1/2 (| 1.1> + | 0,0> + | 1,0> + | 0,1>) = 1/2 ^1/2 (Φ ₊ ²¹ + Ψ ₊ ²¹ ). (18)

The state formally equivalent to Ψ ₁₈ with respect to the operator U _B2B1 is then given by Ψ ₁₉ = U _B2B1 (Ψ ₁₈ ) = 1/2 ^1/2 (| 0,0> + | 1,0>). (19)

If a measurement is carried out on the formally equivalent quantum system in state Ψ ₁₉ , then in 50% of the cases the measurement result "ion 1 and ion 2 do not fluoresce" and in 50% of the cases the measurement result "ion 1 does not fluoresce and ion 2 fluoresces ". The state can be written as a linear combination of the basis vectors Φ ₊ ²¹ and Ψ ₊ ²¹ and the absolute square of the probability amplitudes is equal to 1/2 "over the formally equivalent state Ψ ₁₉ to the quantum system. These already in Eq. However, (18) a recognizable property can not be understood as an inherent property of the quantum system in state Ψ ₁₈ , since the quantum system has the energetic representations associated with the basis vectors Φ ₊ ²¹ and Ψ ₊ ²¹ . The quantum system thus has no inherent properties regarding the state description in the Bell base, but only physically realized properties. This is a remarkable result, bearing in mind that state Ψ _{18 is} a product state and therefore the two systems are considered in the literature to be physically isolated systems (see also the remarks below)!

This connection was not recognized, for example, in [49]. If in [49] this is referred to in connection with the interpretation (IN5) of "more or less isolated systems", the correct formulation introduced below would have to be "more or less energetically isolated systems". Also, [49] in Postulate (P1) did not consider this relationship. As a consequence, two cases must be considered in the thought experiment given in [49] in Section VI. I will also discuss these points in detail below.

However, these examples also reveal a very fundamental difficulty with regard to the interpretation of quantum physics. Looking at quantum systems that are described by a product state, there are no metrological indications that they can not be considered as physically isolated and therefore as independent systems. In contrast, entangled systems show properties that are not possible for independent systems. As an indication that these properties can not be attributed to interactions between the systems involved, the experiments described in [20] and [21] can be considered for teleportation. I will discuss these in more detail below. This then justifies the assumption that entangled systems have properties that can only be considered as a property of the entire system, but not as the sum of the properties of the subsystems. One could therefore argue that entangled quantum systems have "inner properties". In this sense, however, a quantum system in state Ψ ₁₈ also possesses internal properties, since the quantum system has the ones associated with the basis vectors Φ ₊ ²¹ and Φ ₊ ²¹ possess energetic representations and these must therefore be regarded as a physically realized property of the quantum system. Of course, one could argue formally that the properties of a state (vector) are not changed by a base change. Then, however, one must also assign to the state Ψ ₁₆ , the properties that are accessible to the Bell base, physically unrealized, inherent properties, as internal properties. If these considerations are correct, then not only entangled quantum systems but also quantum systems described by a product state possess internal properties. What does mean, however, that every quantum system must have internal properties.

If these considerations are correct, then there are no isolated quantum systems. However, one can say that quantum systems can be differentially "energetically isolated". By this I mean the following: According to the interpretation approach developed above, information about a quantum system is only accessible via the energy representations possible for this quantum system and the quantum systems that are formally equivalent to this quantum system. However, the internal properties of a quantum system are always only indirectly accessible via the computing basis. These are directly accessible only via the quantum systems that are formally equivalent to the quantum system. Looking at the maximum entangled Bell states, they have exactly two energy representations in the calculation base. Of course, the quantum systems that are formally equivalent to these with respect to the operator U _B2B1 have exactly one energy representation. This situation could now be interpreted as meaning that these quantum systems, with regard to the Bell basis, can be assigned exactly one inner property. The quantum systems can thus realize only the energetic representations possible for this inner property. Since the intrinsic properties of a quantum system can be considered as properties of the entire system, it is obvious that the energetic properties of these systems described by the energetic representations should also be considered as properties of the whole system. The Bell states could therefore be described in this sense as energetically not isolated. On the other hand, if one considers a quantum system that can be described by a product state in which the two systems involved are in an energetic eigenstate (as in the case of the state | 1,1>), the states have exactly one energetic representation. However, the formally equivalent quantum systems with respect to the operator U _B2B1 then have two energy representations. However, this could then be interpreted in such a way that the quantum system has to be assigned two internal properties. However, these can not be understood as physically realized properties, since the energetic representations attributed to the internal properties do not correspond to those which can be realized by the quantum system. These systems could therefore be described as energetically isolated. The same result is obtained for product states in which a quantum system is in an energetic eigenstate and one in a superposition state of the form 1/2 ^1/2 (| 0> + e ^iφ | 1>), where with φ any (real) Phase is called. These systems can then be described as energetically isolated. A completely different situation is for example in product states of the form Ψ ₂₀ = e ^iθ / 4 ^1/2 (| 0> ₂ + e ^iφ | 1> ₂ ) (| 0> ₁ ^{-e iφ} | 1> ₁ ), = e ^iθ / 2 ^1/2 (1/2 ^{1 / 2} (| 0,0> - e ^i2φ | 1,1>) + e ^iφ Ψ _- ²¹ ), (20) with: e ^iθ = 1 or e ^{i 2φ} = -1 and the (real) phases θ and φ,
or the product condition Ψ ₁₈ . These always have four energy representations. The quantum systems formally equivalent to these with respect to the operator U _B2B1 always have exactly two energetic representations, as can easily be verified. Thus, these quantum systems have exactly two internal properties with respect to the Bell basis. However, since the energy representations possible for the quantum systems correspond exactly to those assigned to the intrinsic properties of the quantum systems, the intrinsic properties of these quantum systems must be regarded as physically realized properties. However, since the intrinsic properties of a quantum system can not be regarded as properties attributable locally to the subsystems, but only as properties of the total system, the energetic properties of these quantum systems described by the energetic representations must also be considered as properties of the total system. Thus, these quantum systems can not be considered as energetically isolated.

This property of product states of the form Ψ ₂₀ is of very fundamental importance. As shown in Section VII, product states of the form Ψ ₂₀ open up the possibility of assigning state transformations to the requirements of the spin statistics theorem (see, for example, [45]) with regard to the symmetry properties of states for indistinguishable quantum systems, and these as physical realize feasible preparation steps.

For the quantum systems considered above, one could simply introduce the quotient "number of possible energetic representations of the quantum system" / "number of possible energetic representations of the, quantum system describing the inner properties" = QER as a measure of the degree of energetic isolation of the quantum systems , The fact that the quotient QER can be regarded as a measure of the degree of energetic isolation of the states discussed above can be seen from the fact that for the energetically not isolated systems, the maximum entangled Bell states, the product state Ψ ₁₈ and the product states of the form Ψ ₂₀ , QER assumes the maximum possible value QER = 2 and for the energetically isolated product states the smallest possible value QER = 1/2. However, the QER quotient may only be regarded as a sketch to clarify the basic idea for such a definition. Of course, a universally valid definition must be applicable to arbitrary states and not limited only to the cases discussed above as examples and the basis (B2). However, I would not like to elaborate on the question of how to introduce a measure of the degree of energetic isolation of the quantum system in general, since this is irrelevant to the following considerations.

Obviously, in quantum physics, not only does one have to abandon the classical notion that systems can be considered as independent systems if and only if there is no interaction between them, but also distinguish between independent systems and isolated systems. I would therefore suggest the following interpretation:
IN5): There are no isolated quantum systems. The properties assigned to a quantum system by the respective state merely describe the energetic isolation of the quantum system which is more or less strong by the preparation method used. The preparation method used thus limits only the properties attributable to the quantum system. The attributes attributed to the state via the quantum system are actually present. However, these can not be understood in the sense of a physical interaction. The effect of these properties is limited to the energetic representations that can be realized on the quantum system and the formally equivalent quantum systems possible for this quantum system. Only through energetic representations do they become physically accessible through energetic interactions. The energetic representations are thus not carriers of these properties, but merely mediators of these properties.

In this sense, entangled systems are not generated, but correspond only less strongly energetically isolated systems. In the case of physically independently prepared subsystems, this fact is reflected in the inherent properties assigned to them. However, the degree of energetic isolation of a quantum system is not only influenced by the preparation steps used. Of course, even if a measurement is made on a quantum system, it naturally has an influence on the degree of energetic isolation of the quantum system. This influence is reflected in the postulated projection postulate in quantum physics. However, this interpretation already contains the key message for an idea that distinguishes quantum physics. These could be formulated as follows:
IN6): In nature there are properties that are in principle not accessible via physical interactions. One could call these properties internal properties of quantum systems. The intrinsic properties physically attributable to a quantum system are determined by the state that is assigned to the quantum system under consideration in quantum physics. Only the effects of these properties on the considered quantum systems are accessible. The effect of these properties is limited to the fact that these are used to establish realizable energy representations for the quantum system and the formally equivalent quantum systems possible for this quantum system. Only through the actually realized energetic representations are these properties accessible via physical interactions.

As the example for teleportation discussed below shows, it is not absolutely necessary for the participating subsystems to interact physically so that the inherent properties associated with the state of the overall system become accessible via their associated energy representations. I would like to briefly discuss the experiments for teleportation described in [20] and [21]. For these experiments, in the in stored linear ion trap three ions stored. The goal of these experiments is to transfer the state of one qubit (Ion 1) to another qubit (Ion 3) without any direct physical interaction between these two qubits at any one time. For this purpose, in a first step, the ions 3 and 2 (the left ion and the middle ion) are prepared in the Bell state Ψ ₊ ³² . Thereafter, the ion ^{1 is} prepared in any state Ψ _from ¹ . For the success of the experiment, it is not necessary to know this condition. The state for the entire system is then given by: Ψ _T = 1 / (2) ^1/2 (| 1> ₃ | 0> ₂ + | 0> ₃ | 1> ₂ ) Ψ _ab ¹ (21) With Ψ _from ¹ = 1 / (2) ^1/2 (a | 0> ₁ + b | 1> ₁ ); | A | + | b | ² = 1. (22)

Next follows Ψ _T = 1 / (4) ^1/2 (a | ¹ > ₃ | 0> ₂ | 0> ₁ + b | 1> ₃ | 0> ₂ | 1> ₁ + a | 0> ₃ | 1> ₂ | 0> ₁ + b | 0> ₃ | 1> ₂ | 1> ₁ ) = 1 / (8) ^1/2 ((a | 1> ₃ + b | 0> ₃ ) Φ ₊ ²¹ + (a | 1> ₃ - b | 0> ₃ ) Φ _- ²¹ + (a | 0> ₃ + b | 1> ₃ ) Ψ ₊ ²¹ + (a | 0> ₃ - b | 1> ₃ ) Ψ _- ²¹ ) = 1 / ( 4) ^1/2 ((XY _from ³ ) Φ ₊ ²¹ + (XZΨ _from ³ ) Φ _- ²¹ + Ψ _from ³ Ψ ₊ ²¹ + (ZΨ _from ³ ) Ψ _- ²¹ ) (23) where used | 0> ₂ | a> ₁ = 1 / (2) ^1/2 (Φ ₊ ²¹ + Φ _- ²¹ ) | 1> ₂ | 1> ₁ = 1 / (2) ^1/2 (Φ ₊ ²¹ - Φ _- ²¹ ) | 1> ₂ | 0> ₁ = 1 / (2) ^1/2 (Ψ ₊ ²¹ + Ψ _- ²¹ ) | 0> ₂ | 1> ₁ = 1 / (2) ^1/2 (Ψ ₊ ²¹ - Ψ _- ²¹ ) and 1 / (2) ^1/2 (a | 0> ₃ + b | 1> ₃ ) = Ψ _from ³ 1 / (2) ^1/2 (a | 1> ₃ + b | 0> ₃ ) = XΨ _from ³ 1 / (2) ^1/2 (a | 1> ₃ - b | 0> ₃ ) = XZΨ _from ³ 1 / (2) ^1/2 (a | 0> ₃ - b | 1> ₃ ) = ZΨ _from ³ with the Pauli operators X and Z

If one _applies the operator U _B2B1 , which acts only on ion 2 and ion 1, to the state Ψ _T , then this becomes the state Ψ _TB2B1 = 1 / (4) ^1/2 ((XΨ _from ³ ) | 0,0> + (XZΨ _from ³ ) | 0,1> + Ψ _from ³ | 1,0> + (ZΨ _from ³ ) | 1 , 1>) (24) transferred. The state Ψ _TB2B1 is thus formally equivalent to the state Ψ _T. If a measurement is now carried out on the ions 2 and 1, the state Ψ _from ¹ into which the ion 1 was originally prepared is now accessible via ion 3. If, for example, the measurement result "Ion 1 fluoresces and Ion 2 does not fluoresce, then it is known that the ion 3 must then be in the state Ψ _ab ³ . In other cases, the ion 3 is not after a measurement of the ion 1 and 2 while in the state Ψ _from ^3, but may be converted always therein. Is obtained, for example, the measurement result "Ion fluoresces 1 and 2 fluoresces Ion", one needs to apply in order to convert this into the state Ψ _from ³ only the operator X ^-1 on the ion. 3 Where X- ^{1 is} the operator inverse to X. In order to be able to analyze the state of ion 3, it is therefore absolutely necessary to perform a measurement on ions 1 and 2 before the state analysis on ion 3. However, as measured values are always classical quantities, a classical information channel is also required for the teleportation experiments described in [20] and [21] in order to be able to carry out the state analysis on Ion 3.

• IV. Elementare Quantensysteme: Da nach den Interpretationen (IN5) und (IN6) die einem Quantensystem, über dessen Zustand zugeordneten, Eigenschaften zwar durch die energetischen Repräsentationen physikalisch zugänglich werden, die energetischen Repräsentationen aber nicht als Träger dieser Eigenschaften betrachtet werden können, liegt die Vermutung nahe, dass es auch Quantensysteme geben könnte, die keine energetische Repräsentation besitzen. Dass es solche Quantensysteme gibt, möchte ich anhand der folgenden Überlegungen aufzeigen.

In order to be able to interpret the experiments, it is necessary to take a closer look at the individual steps. In the first step, ions 3 and 2 are prepared in the Bell state Ψ ₊ ³² . In this step, there is no physical interaction with the ion 1. Then, the ion ^{1 is} prepared in the state Ψ _from ¹ . Also in this step the ion 1 does not interact with the ions 3 and 2. Thus one has two Physically prepared completely independent systems. The only way to assign a state within quantum physics to a whole system consisting of two physically independent systems is to associate with it the product state formed over the product of the states of the individual systems. Thus, the total system consisting of the three ions is in the state Ψ _T Gl. (21). But the quantum system can now also be solved by Eq. (23) properties. It is precisely these properties that enable teleportation. However, the quantum system has these properties only as an inherent property. In order to make these metrologically accessible, the corresponding energetic representations must be realized on the quantum system. This is achieved by the fact that the state Ψ _{T is converted} by means of the operator U _B2B1 in the formally equivalent state Ψ _TB2B1 . Only then are those in Eq. (23) described in the arithmetic base of the quantum system in terms of metrology accessible via the corresponding energetic representations. For it is only in the state Ψ _TB2B1 that one can speak of four fundamentally indistinguishable possibilities in terms of the superposition principle, since only in this case is the unmistakable property of the quantum system necessary for this given by the energy stored in ions 1 and 2. If, for example, the measurement result "Ion 2 fluoresces and Ion 1 does not fluoresce" then, for example, when measuring at the ions 1 and 2, then the quantum system must be present after the measurement at the ions 1 and 2 in the state Ψ _from ³ | 1.4> ,

• IV. Elementary quantum systems: Since according to the interpretations (IN5) and (IN6), the properties assigned to a quantum system, whose properties are physically accessible through the energetic representations, can not be considered to carry the energy representations Conjecture that there may also be quantum systems that have no energetic representation. I want to show that there are such quantum systems based on the following considerations.

If two ions are prepared in the Bell state Ψ _- ²¹ and then the two ions are illuminated with a suitable π-pulse [17], the state Ψ _- ^{21 can change} to the state = ₂₅ = 1/2 ^1/2 (| 1,2> - | 2,1>) (25) be transferred. The two ions are then in the electronic ground state (4 ² S _1/2 ). Since the nuclear spin of the ⁴⁰ Ca ⁺ ions used equals zero, the state Ψ _{25 describes} a system composed of two spin ½ systems with total spin 0. The electronic ground state is transformed into the energetically lower state by the Zeemann effect [22] 1> (ground state of the qubit) with the magnetic quantum number m = -1/2 and the higher energy state | 2> (excited state of the qubit) with the magnetic quantum number m = +1/2. If the magnetic field superimposed in the z-direction of the ion trap is switched off, the two energy levels | 1> and | 2> coincide. In this process, the spin degree of freedom of one of the two ions must then give the energy ΔE _z / 2 and the spin degree of freedom of the other ion absorb the energy ΔE _z / 2, where ΔE _{z is} the energy difference between the two states | 2> and | 1> when the magnetic field is activated. However, one can not decide in principle whether the spin degree of freedom of the left or right ion absorbs or gives off the energy ΔE _z / 2. Since the magnetic field superimposed on the ion trap changes over time during the switch-off process, the energy ΔE _z / 2 could be exchanged via the magnetic field. however, in sum, the magnetic field does not absorb energy from the system's spin-degree of freedom, nor does it summon energy to the spin-degree of freedom of the system. The very analogous process occurs when the magnetic field is switched on again. In the ideal case, after the magnetic field has been switched on again, the quantum system is again present in the energetically represented state Ψ ₂₅ . However, as long as the magnetic field is switched off, no energetic representation can be assigned to the system in state Ψ ₂₅ , since the two energy levels | 1> and | 2> coincide energetically. It is then not possible, in the sense of the superposition principle, to speak of basically indistinguishable possibilities, since the characteristic property required for this via the energetic representations is not given. I would like to refer to quantum systems that do not possess energetic representations as "elementary quantum systems". I would like to call the state associated with an elementary quantum system "elementary state".

By means of the spin-degree of freedom, it is thus possible in a simple manner to prepare elementary quantum systems. For a better understanding, let me briefly describe some properties of systems with spin ½. For this purpose, it is assumed below that the magnetic moment μ associated with the spin ½ system is aligned anti-parallel to the spin. This does not limit the generality of the statements made, since this assumption is only needed in order to describe the interaction of a spin ½ system with magnetic fields as simply as possible. With μ is also the amount of maximum value of the z-component of μ = (μ _x , μ _y , μ _z ) denotes. Since there is no likelihood of confusion, μ is also referred to below as the magnetic moment.

If a system with spin ½ exists in a static magnetic field B _z , with the quantization axis z '(the magnetic field points in the direction z'), then the spin degree of freedom can be determined by the interaction energy W = -μB, (26) with B = (0, 0, B _{z '} ) and B _z' > 0, two energy eigenstates are assigned with respect to the quantization axis z given by the magnetic field [22]. The energetically higher state | + z '> with the interaction energy W = + μB _z' at which the spin is aligned parallel to the magnetic field (m = 1/2) and the lower energy state | -z '> with the interaction energy W = -ΜB _{z '} where the spin is anti-parallel to the magnetic field (m = -1/2). The energy difference ΔE _{z 'of} the two states is then calculated to ΔE _{z '} = f _Lam h = 2μB _z' (27) where f _Lam denotes the Lamor frequency. If the magnetic field is not homogeneous but has a gradient, such as the magnetic field in a Stern-Gerlach magnet, the direction in which the system is accelerated depends on the interaction energy W = -μB and on the gradient of the magnetic field [23]. In the following, I assume that the systems under consideration carry no charge. If the magnetic field B has the gradient dB _{z '} / dz'<0, then the system is accelerated in the direction + z 'when W = + μB _z' and accelerates in the direction -z 'when W = -μB _{z '} is. If the system is in a superposition state with respect to the quantization axis z ' Ψ ₂₈ = a ₊ | + z '> + a _- | -z'>, (28) thus it can not be predicted in which direction (+ z 'or -z') the system is accelerated in the Stern-Gerlach magnet. It can therefor only a probability that over the square value of the probability amplitudes a ₊ and a _- is set to be given [23]. The Stern-Gerlach magnet is named after W. Gerlach, O. Stern, who was the first to experimentally prove the direction quantization of being [24], [25].

If a spin ½ system enters a static, here homogeneous magnetic field (see ), the longitudinal Stern-Gerlach effect [26] occurs. As long as the system is in field-free space, its total energy E _ges = E _{0 is} equal to the kinetic energy E _kin = ½mu ² , where m is the mass of the system and u is the velocity of the system. If the system enters the magnetic field B _{z '} with the quantization axis z', the transition from the field-free region into the magnetic field can be considered as if the system moves into a potential. Within the magnetic field B _{z '} , the energy associated with the spin degree of freedom via the interaction energy W is again given for a spin ½ system in the state | + z'> equal to E _spin = + μB _{z '} and for a spin ½ system in the state | - z '> equal to E _spin = -μB _z' . In the field-free space Espin = 0, of course. With the entry of the system into the potential, the energy levels of the spin-degree of freedom do not coincide, but are raised by a value ΔE _{z '} / 2 for a system in the state | + z'> and for a system in the state | -z '> lowered by the value ΔE _z' / 2. With respect to the field-free case in which no energy can be assigned to the spin degree of freedom, the system entering the potential must be within the time specified by the energy-time uncertainty principle Δt ≤ h / (4πΔE _{z '} / 2) (29) absorb or release the corresponding energy, otherwise the law of energy conservation would be violated. But now the system can not exchange energy with the static magnetic field. Thus, the energy exchange can only take place within the system. So the energy has to be exchanged with the kinetic energy of the system. If the system is in the state | + z '>, the total energy continues to be E _ges = E ₀ . The kinetic energy is now E _kin = E ₀ - ΔE _{z '} / 2. If the system is in the state | -z '>, then the total energy continues to be E _ges = E ₀ . The kinetic energy is now E _kin = E ₀ + ΔE _{z '} / 2. After the system has traversed the transition region between the field-free region outside the magnet and the homogeneous region of the magnet, the system flies in state | + z '> at a velocity <u through the homogeneous region of the magnetic field and the system in state | -z '> at a speed> u through the homogeneous part of the magnetic field. When exiting the magnetic field the process described reverses and in both cases the systems continue to fly at the speed u.

An energetic shift of the energy levels | + z '> and | - z'> thus not only occurs when a magnetic field is switched on or off, as in the example above, but also occurs when a spin ½ system into a static Magnetic field occurs. For spin ½ systems, the unique property required with respect to the superposition principle is always given by the interaction energy W = -μB. The spin-degree of freedom can therefore be assigned no energetic representation in field-free space. If a magnetic field is then switched on, or if the systems enter a static magnetic field, then this process can also be understood as a preparation step, in which the elementary quantum system is converted into a corresponding energetically represented quantum system. Unless one considers that the elementary quantum system already determines whether the quantum system takes up or releases energy in the spin degree of freedom in this preparation step, but this would mean that one would introduce a hidden variable, this becomes clear only with the preparation step determine at random whether the quantum system absorbs or gives off energy in the spin degree of freedom if several energetic representations are possible. Similarly, the reverse process can be interpreted. In this case, the corresponding elementary quantum system is then prepared on the basis of an energetically represented quantum system.

To get information about the spin degree of freedom, you need a suitable measuring device, a spin analyzer (SA). For this purpose, a star Gerlach magnet can be used in the simplest case. The information of the quantum system accessible via a Stern-Gerlach magnet having a magnetic field B _{z '} with the quantization axis z' and the gradient dB _{z '} / dz'<0 is then given in which direction (+ z 'or -z' ) the spin ½ systems are deflected. As a result, one can obtain "The spin ½ system was deflected in the + z 'direction" or "The spin ½ system was deflected in the -z'direction". Since individual measurement results are determined by the respectively realized energy representation and thus by the interaction energy W, these can be interpreted as follows:
IN7): The measurement result "The spin ½ system is deflected in the direction + z 'represents the information that the spin degree of freedom of the spin ½ system with entry into the magnetic field B _z' , certainly the energy ΔE _{z '} / 2 = μB _{z 'has} recorded. The measurement result "The spin ½ system is deflected in the direction -z 'represents the information that the spin degree of freedom of the spin ½ system with entry into the magnetic field B _z' , certainly the energy ΔE _{z '} / 2 = μB _{z 'has} delivered.

For a spin ½ system in the state | + z '> (| -z'>) one can then give the following interpretation with respect to the quantization axis z ', analogous to (IN2):
IN8): If a spin ½ system is in the state | + z '> it represents the information that the spin-degree of freedom has to absorb the energy ΔE _z' / 2 = μB _{z '} as soon as the system enters the magnetic field B _z'. entry. If a spin ½ system is in the state | -z '> it represents the information that the spin-degree of freedom has to give off the energy ΔE _z' / 2 = μB _{z '} as soon as the system enters the magnetic field B _z' .

If there is a spin ½ system with respect to the quantization axis z 'in a superposition state Ψ ₂₈ Eq. (28), this can be assigned to the interpretation (IN9) analogously to (IN3).
IN9): A superposition state Ψ = a ₊ | + z '> + a _- | -z'> represents the information that there are two ½ basically indistinguishable possibilities: 1.): "The spin-degree of freedom of the spin system has to enter into the magnetic field B _{z '} the energy ΔE _z' / 2 = μB _{z '} or 2.): "The spin degree of freedom of the spin ½ systemss has the energy ΔE _z' / 2 = with entry into the magnetic field B _{z '} μB _{z '} delivered'. The probability of the two possibilities is given by the sum of the sums of the amplitudes a ₊ and a _- .

Even quantum systems composed of several spin ½ systems, one can assign an interpretation in this way. For example, is a quantum axis z 'in the Bell state consisting of two spin ½ systems Ψ ₃₀ = 1/2 ^1/2 (| -z ', + z'> - + z ', -z'>) (30) and B _z'L = B _z'R = B _{z '} , where B _z'L and. B _z'R the magnetic fields are designated in which the subsystems S _L (left subsystem) or SR (right subsystem), with | S _L , S _R > = | S _L > | SR>, can occur (see also ), this can be assigned to the interpretation (IN10) analogously to (IN4).
IN10): The Bell state Ψ ₃₀ represents the information that there are two basically indistinguishable possibilities: 1.): "The spin degree of freedom of the spin ½ system S _L takes the energy ΔE _{z when} entering a magnetic field B _{z'L '} / 2 = μB _z' and the spin degree of freedom of the spin ½ system SR gives with entry into a magnetic field B _z'R the energy ΔE _{z '} / 2 = μB _z' ab "or 2.):" The spin degree of freedom of the spin ½ system S _L gives the energy ΔE _{z '} / 2 _when entering a magnetic field B _z'L = μB _{z '} and the spin degree of freedom of the spin ½ system SR takes on the energy ΔE _z' / 2 = μB _{z ' when} entering a magnetic field B _z'R ".

In the previous considerations, the spatial part of the state was always suppressed and only the spin share is taken into account. This is justified as long as the spin-degree of freedom preparation steps do not change the spatial state component and the spatial state component is irrelevant to understanding the considerations. The only effect that will be used in the following for state preparation and that could change the spatial state component is the longitudinal Stern-Gerlach effect. The longitudinal Stern-Gerlach effect has, for example, an influence on the spatial state component if the quantum system with respect to the quantization axis z 'of the magnetic field B _{z' is} in a superposition state Ψ ₂₈ Eq. (28). In this case, the components forming the state propagate at different speeds in the magnetic field [27]. As a result, the spatial units associated wave trains (de Broglie wave packets) are shifted against each other in the propagation direction. Only if this shift remains small compared to the extent of the wave packet and thus small compared to the coherence length, this influence can be neglected. Since this can always be ensured in the following, I still do not want to explicitly state the spatial state part.

Since the following considerations refer to the EPR-Bohm thought experiment [28], I would like to briefly describe this here and show how this can be interpreted by means of the concepts introduced above elementary quantum system and energetically represented quantum system. The thought experiment was first proposed by David Bohr [29]. The essential idea of the EPR-Bohm thought experiment is the following: In a source Q, a system that is ideally at rest before decay is split with total spin 0 into two indistinguishable subsystems with spin ½, preserving the overall spin such that the two subsystems look classically on a single spin straight line along the x axis in the opposite direction with the same velocity u flying away from each other ( ). If the subsystem SR flying to the right along the x-axis is conventionally detected at any distance A _R from the source Q, then it is certain that at the same time it can detect the left-flying subsystem S _L at the same distance, since the entire system has decayed into two subsystems while preserving the total momentum. In quantum physics, however, two complementary quantities [30] can not be determined simultaneously with arbitrary precision. The more accurately one locates a particle (the smaller the extent of the source is) the more indefinite the momentum of the particle becomes. Since it is sufficient for the EPR-Bohm thought experiment and for further discussion, if the source allows localization of the particles in the range of some order and the boundary conditions can always be chosen in the following, that by the localization of the particles in the source caused impulse blur can be neglected, the movement of the particles can be treated classically. The two along the x-axis in the distance A _R right and A _L left of the source Q arranged identical star Gerlach magnets M _R and M _L are used for state analysis. The quantization axes are denoted by z ' _R and z' _L , respectively. The two quantization axes are always below in the xy plane.

If the two spin ½ subsystems forming the entire system are indistinguishable, then the quantum physical state describing the entire system must be antisymmetric due to the Pauli principle and the two spin ½ systems interchanged. This results in the description of the overall system immediately the state = ₃₁ = 1/2 ^1/2 (| -, +> - | +, ->), (31) where "+" means that the corresponding subsystem is assigned the spin + ½ and "-" means that this subsystem is assigned the spin -½, with | S _L , S _R > = | S _L > | S _R >. An essential property of the maximally entangled state Ψ ₃₁ thus prepared is that no quantization direction can be assigned to it [28], [29], [31]. It is therefore not possible to interpret the state Ψ ₃₁ thus prepared in such a way that, although it has a direction of quantization, it is not known.

If A _L = A _R and for example B _z'L = B _z'R = B _{z '} with z' _L = z ' _R = z', the elementary state Ψ ₃₁ Eq. (31) in the energetically represented state Ψ ₃₀ Eq. (30) transferred as soon as the subsystems enter the respective magnetic fields. It does not matter in which direction the quantization axis z 'points. With regard to the quantization axis z ', one always obtains the energetically represented state Ψ ₃₀ . Thus depend the possible for the quantum system energetic Representations, in contrast to all previously considered quantum systems, not from the quantization direction z 'from. In accordance with the actually realized energy representations in the spin degree of freedom, the subsystem S _L (S _R ) is then deflected either in the direction + z '(-z') or in the direction -z '(+ z'). The spatial components (de Broglie wave packets) assigned to the two state components of the state Ψ ₃₀ are thus so far separated from one another that they no longer spatially overlap and the actually realized energetic representations lead to distinguishable events on the detectors D _L and D _R , As soon as the two subsystems S _L and S _{R have left} the respective magnetic fields, however, no energy representation can be assigned to the quantum system with regard to the degree of spin-freedom. Nevertheless, there is an energetically represented quantum system. Energetic representations can now be assigned to the quantum system in the sense of the superposition principle by the possible paths which the individual subsystems can take. The Stern-Gerlach magnets thus transform the originally elementary quantum system into a quantum system energetically represented with respect to the spatial degree of freedom. As long as it is not even possible in principle to decide in which direction the subsystems S _L and S _R were deflected in the respective Stern-Gerlach magnets, the quantum system then entangled with respect to the spatial degree of freedom becomes the two possible energetic representations 1. ) "The kinetic energy of the subsystem S _L is localized in the wave packet propagating in the direction + z 'and the kinetic energy of the subsystem S _R is located in the wave packet propagating in the direction -z'" or 2.) "The Kinetic energy of the subsystem S _L is located in the wave packet propagating in the -z 'direction and the kinetic energy of the subsystem S _R is localized in the wave packet traveling in the + z' direction. The elementary quantum system, which was originally maximally entangled with regard to the degree of spin-freedom, is thus transferred with increasing separation of the spatial state components into a maximally entangled quantum system which is energetically represented with regard to the spatial degree of freedom. Once the spatial state parts no longer overlap, the entanglement has then been completely transferred from the spin degree of freedom to the spatial degree of freedom. As soon as one knows, if only in principle, that one could know in which direction the subsystems S _L or S _{R were} deflected, the state Ψ ₃₀ either becomes the state | + z ' _L , -z' _R > or in the state | -z ' _L , + z' _R > transferred.

• V. Kohärent gekoppelte Vakuumfluktuationen: Nach den bisherigen Überlegungen kann man die Überführung eines elementaren Quantensystems in das entsprechende energetisch repräsentierte Quantensystem als Präparationsschritt begreifen. Dieser Präparationsschritt findet immer dann statt, wenn ein Magnetfeld eingeschaltet wird, oder das betreffende Quantensystem in ein Magnetfeld eintritt. Tritt beispielsweise ein Spin ½ System im Zustand |+z'> in ein Magnetfeld B_z' mit der Quantisierungsachse z' ein, so weiß man mit Sicherheit, dass dem System dann die energetische Repräsentation „Der Spin-Freiheitsgrad hat die über die Wechselwirkungsenergie W festgelegte Energie E_spin = +μB_z' (das System nimmt die Energie E_spin = +μB_z' im Spinfreiheitsgrad auf, sobald das System in das Magnetfeld eintritt)” zugeordnet werden kann. Das gleiche gilt, wenn ein Magnetfeld eingeschaltet wird. Befindet sich das System im Zustand |+z'> und wird dann ein Magnetfeld B_z' mit der Quantisierungsachse z' eingeschaltet, so weiß man mit Sicherheit, dass dem System dann die energetische Repräsentation „Der Spin-Freiheitsgrad hat die über die Wechselwirkungsenergie W festgelegte Energie E_spin = +μB_z' (das System nimmt die Energie E_spin = +μB_z' im Spinfreiheitsgrad auf, sobald das Magnetfeld eingeschaltet wird)” zugeordnet werden kann. Wenn man aber die Überführung eines elementaren Quantensystems in das entsprechende energetisch repräsentierte Quantensystem als Präparationsschritt auffassen kann, so stellt sich die Frage, ob es dann nicht auch noch andere Präparationsschritte zur Überführung eines elementaren Quantensystems in ein energetisch repräsentiertes Quantensystem geben könnte, um die für das elementare Quantensystem möglichen energetischen Repräsentationen zu realisieren? Was dann aber die Frage aufwirft, auf welche Art und Weise der Spin-Freiheitsgrad sonst noch Energie austauschen könnte?

If A _L > A _R and the right-flying subsystem S _R enters the magnetic field of the Stern-Gerlach magnet arranged at a distance A _R from the source Q, then, as soon as one knows, if only that one is in the Principle could know in which direction the subsystem is deflected, the state of the system either in the state | + z ' _L , -z' _R > or in the state | - z ' _L , + z' _R > transferred. If the two quantization axes of the Stern-Gerlach magnets used are parallel, then the two possible measurement results are always strictly anti-correlated, regardless of the direction of the quantization axes. If the subsystem S _R in the direction of + z deflected _'R (z' _R), then one knows with certainty that the subsystem S _L z in the direction _'L (+ z' _L) is deflected is now the quantization of the left-ordered by the source at a distance A _L Stern-Gerlach magnetic not parallel to z _'R, so there is the left flying subsystem S _L, as soon as it enters the magnetic field of the star-Gerlach magnet with respect to z' _L in a superposition state and the two measurement results are then no longer strictly anti-correlated. If, for example, the two quantization axes z ' _L and z' _R are perpendicular to one another, no correlation can be established between the two measurement results. The EPR-Bohm thought experiment can thus also be interpreted in such a way that a quantization direction (here z ' _R ) is established for the system when the right subsystem enters the magnetic field of the Stern-Gerlach magnet. First, if the left-flying subsystem (A _L <A _R ) enters the magnetic field of the Stern-Gerlach magnet located to the left of the source at a distance A _L , the quantization direction of the system is set to z ' _L by this magnet. Regardless of whether A _L> A _R A _L <A _R or A _L = A _R is selected, the two measured values are thus always strongly anti-correlated, as long as the two quantization z _'L and z' _R are parallel.

• V. Coherently Coupled Vacuum Fluctuations: According to previous considerations, the transformation of an elementary quantum system into the corresponding energetically represented quantum system can be understood as a preparation step. This preparation step always takes place when a magnetic field is switched on, or the relevant quantum system enters a magnetic field. If, for example, a spin ½ system in the state | + z '> enters a magnetic field B _z' with the quantization axis z ', then one knows with certainty that the system then has the energy representation "The spin degree of freedom has the interaction energy W fixed energy E _spin = + μB _{z '} (the system takes on the energy E _spin = + μB _z' in the spin degree of freedom as soon as the system enters the magnetic field) ". The same is true when a magnetic field is turned on. If the system is in the state | + z '> and then a magnetic field B _{z' is switched on} with the quantization axis z ', then one knows with certainty that the system then has the energetic representation "The spin degree of freedom has the interaction energy W fixed energy E _spin = + μB _{z '} (the system takes on the energy E _spin = + μB _z' in the spin degree of freedom, as soon as the magnetic field is turned on) "can be assigned. If However, if the transfer of an elementary quantum system into the corresponding energetically represented quantum system can be considered as a preparation step, then the question arises, if there could not be other preparation steps for the transformation of an elementary quantum system into an energetically represented quantum system, that for the elementary quantum system to realize possible energetic representations? But then what raises the question, in what way the spin-degree of freedom could otherwise exchange energy?

Now, for the vacuum about the energy-time uncertainty relation Eq. (29) given time .DELTA.t the energy .DELTA.E _{z '} / 2 = μB _z' are removed by virtual photons. These vacuum fluctuations are responsible for a whole series of physical phenomena [33]. One phenomenon is the Casimir effect. The Casimir effect can be understood as a radiation pressure caused by the virtual photons [34]. If, for example, two plane-parallel mirrors are opposed to one another, they will attract each other due to the radiation pressure of the virtual photons which is slightly reduced between the two mirrors compared to the outer space. The resulting force is called Casimir force. The Casimir force can now be measured with high accuracy [35]. Now, however, the system (the two mirrors) can be depleted of energy as they move towards each other. As long as you do not pull the two mirrors apart again, you have to extract this energy from the vacuum. If you do not want to give up the energy conservation law, you should expect the vacuum to pick up the same energy again at some point, otherwise the energy conservation law would be violated. The question then arises as to whether there is a finite propagation velocity v _c ≤ c for this energy exchange "in vacuum", or whether a local "extraction of energy from the vacuum" takes place instantaneously at another, possibly far away, site "Energy release to the vacuum" can make noticeable? I would like to go into this question in more detail below and assume here that this energy exchange with the finite velocity v _c = c is possible. Going one step further, one comes to the following hypothesis:
The vacuum can be taken locally at a location for an arbitrarily long time, in the form of a photon generated via a vacuum fluctuation, provided the same amount of energy to the vacuum again in the form of a photon at any other location locally while maintaining the energy-time uncertainty principle Eq. (29) is discharged again and thus the vacuum is taken in the sum neither energy removed after.

Now this process will not happen by itself. As with the Casimir effect, special boundary conditions (there the resonator formed by the mirrors) may be required to facilitate this process. One way to produce the necessary conditions that could offer ₃₁ considered in the EPR-Bohm thought experiment system in the state Ψ. Because contact the two subsystems S _L and S _R at the same time in the magnetic fields of the Stern-Gerlach magnet M _L and M _R a, a subsystem needs to (IN9) the energy .DELTA.E _{z '/} 2 = mB _z' in the spin degree of freedom record and the other subsystem gives the energy ΔE _{z '} / 2 = μB _z' in the spin degree of freedom, provided B _z'R = and B _z'R = B _{z '} and z' _R = z ' _L = z'. Which subsystem absorbs energy and gives off energy, however, can not be predicted in principle. This situation could therefore also be interpreted in such a way that it is then determined via the vacuum fluctuations which subsystem receives a virtual photon and which emits a photon to the vacuum. But how could this process actually happen?

For this I would like the in look at the arrangement shown. As in the EPR-Bohm thought experiment, the source Q emits a quantum system composed of two spin ½ systems in state Ψ ₃₁ . The two subsystems again move away from the source Q on a straight line along the x-axis in the opposite direction with the same velocity u. Along the x-axis, the magnets M _L and M _R are arranged at a distance A _{L to the} left of the source and at a distance A _{R to the} right of the source. These have a homogeneous magnetic field with B _zR = B _zL = B _{z '} z _R = z _L = z and A _R = A _L. The region with a homogeneous magnetic field to the magnet M _{L is} the length L _ML and for the magnet M _R have the length L _MR. Of the Transition region from the field-free space in the area with the homogeneous magnetic field B _z should have the length Δs _B. Δt _B = Δs _B / u denotes the time of flight which the subsystems need to fly through the distance Δs _B. Furthermore, the boundary condition (R1) should be fulfilled: Δt _B ≤ Δt _max = h / (4πΔE _z / 2) (R1.1) (A _L + A _R ) / v _c h / (4πΔE _z / 2) (R1.2)

The condition (R1.1) ensures that the energy-time uncertainty equation Eq. (29) maximum predetermined time .DELTA.t _{max is} greater than or equal to the time of flight .DELTA.t _B , which require the subsystems to traverse the transition region from field-free space in the area with the homogeneous magnetic field B _z . The condition (R1.2) ensures that an energy exchange between the two subsystems within the spin degree of freedom is possible in the maximum time Δt _max given by the energy-time uncertainty principle.

If the two subsystems S _L and S _R enter the magnetic fields of the respective magnets, the elementary state Ψ _{31 is converted} into a state which is energetically represented with respect to the quantization axis z. However, as long as the subsystems are still in the respective transitional areas, it can not yet be said that the quantum system has realized an energetic representation, since only for Δt ≥ Δt _vir = (Δt _B + Δt _max ) (32) an energy exchange with the spin degree of freedom is completed with certainty. Eq. (32) is a somewhat generous estimate, but is sufficient for further discussion, as the exact value is not relevant below. However, as soon as the subsystems enter the respective transition region (Δt = 0) and as long as the subsystems in the spin degree of freedom have not yet exchanged any energy, the quantum system can be assigned a "virtual energy representation". The two possible virtual energetic representations are then given by: 1.): "The spin degree of freedom of the subsystem S _L has the virtual energy E _{+ vir} and the spin degree of freedom of the subsystem S _R with entry into the transition region of the magnetic field B _z with entry into the transition region of the magnetic field B _z the virtual energy E _-vir or 2.): "The spin degree of freedom of the subsystem S _L has the virtual energy E _-vir and the spin degree of freedom with entry into the transition region of the magnetic field B _z of the subsystem S _R has with entry into the transition region of the magnetic field B _z the virtual energy E _{+ vir} '', with E _{+ vir} (Δt) = E _{+ spin} (Δt) - μB _z (Δt) and E _-vir (Δt) = E (Δt) + μB _Z (Δt) (33) With: E _{+ spin} (Δt) = μB _z (Δt) and E (Δt) = -μB _z (Δt), where the dependence of the individual quantities on the variable Δt results from the fact that the magnetic field in the transition region is not homogeneous and thus the value of the individual quantities depends on the respective locations of the subsystems and thus on Δt. The in Eq. (33) selected notation is intended to express the following: If a subsystem is assigned the virtual energy E _{+ vir} (Δt) (E _-vir (Δt)), then the subsystem develops the energy representation represented by the energy E _spin = μB _z (E _spin = -μB _z ), provided that the subsystems have the ability to exchange energy between the spin-degree of freedom and the kinetic energy via the longitudinal Stern-Gerlach effect. However, the longitudinal Stern-Gerlach effect can not occur if the virtual energy subsystem E- _{vir interacts} with a virtual photon of energy μB _z and _captures the energy of the virtual photon. Because then this can no longer be assigned to the virtual energy E _-vir , since due to the absorption of the virtual photon the subsystem after the energy E _spin = μB _z must be assigned. The other subsystem with the virtual energy E _{+ vir} must then release the energy μB _z in the form of a photon to the vacuum in the time given by the energy-time-uncertainty relation. This subsystem is thus assigned the energy E _spin = -μB _z . Since under these boundary conditions an energetic representation is always realized for the state Ψ ₃₁ , I would like to describe the absorption of a virtual photon by one subsystem and the delivery of one photon to the vacuum by the other subsystem as coherently coupled vacuum fluctuation necessary for this preparation process , The occurrence of a coherently coupled vacuum fluctuation could therefore also be interpreted as causing a synchronous folding over of the spins of the two subsystems. Eq. (33), however, should not be interpreted in such a way that it first determines which virtual energy representation has been realized and thereafter a coherently coupled vacuum fluctuation occurs. Virtual energetic representations can only be spoken in a formal sense in order to describe the process of a coherently coupled vacuum fluctuation. It can only be said of the energetic representations that one of the possible energetic representations is physically realized via a coherently coupled vacuum fluctuation.

The situation can be described quite analogously if an energetically represented quantum system in the singlet state Ψ _{- is transformed} into an elementary quantum system, by exiting the subsystems from the static magnetic fields or by switching off the magnetic fields. However, this is too Note that the actually realized energetic representation then appears as an additional boundary condition. The actually realized energetic representation then determines which subsystem a photon emits to the vacuum and which picks up a virtual photon from the vacuum.

If these considerations are correct, then for an elementary quantum system in state Ψ _31, the occurrence of a coherently coupled vacuum fluctuation with entry of the subsystems into the respective static magnetic fields can not be distinguished from the case by the resulting energetically represented state if the two magnets M _L and M _{R are} only turned on as soon as the subsystems have passed the transition region of the respective magnets, since in both cases, apart from an unobservable global phase, the state Ψ _{30 which} is energetically represented with respect to the quantization axis z '= z results. As long as both subsystems are in the homogeneous region of the magnets M _L and M _R , the state Ψ ₃₀ remains unchanged since B _zR = B _zL = B _Z. How the temporal evolution of the phase of a spin ½ system can be described by quantum physics is given in [36]. If one wishes to ensure that the longitudinal Stern-Gerlach effect occurs with the exit of the subsystems from the respective magnetic fields, this can be ensured, for example, by determining the lengths L _ML and L _MR according to FIG L _MR > (Δt _vir u + L _ML ) (34) chooses. Since the subsystem S _R is then still within the magnetic field, when the subsystem S _L has already left the magnetic field again, the phases of the two state components now develop differently. It is easy to verify that the quantum system is in the state after both subsystems have left the respective magnetic fields Ψ ₃₅ = 1/2 ^1/2 e ^{-iθ / 2} (| -z, + z> ^{-e iθ} | + z, -z>), (35) With θ = 2πΔE _z Δt _MRL / h and Δt _MRL = (L _MR -L _ML ) / u By choosing ΔE _z and Δt _MRL such that .DELTA.E _z .DELTA.t _MRL / h = 1, (36) Thus, after the two subsystems have left the magnetic fields, the quantum system is again up to the global phase e ^-iπ = -1 in state Ψ ₃₁ . The prerequisite for this, however, is that the longitudinal star Gerlach effect practically does not change the spatial state components. That the boundary conditions can be selected accordingly, I will show below an example.

Of course, as soon as all subsystems are no longer in the magnetic field B _z , it can no longer be said that the entire quantum system is energetically represented. As for the subsystems that are still within the magnetic field B _z which denote this attributable to the spin degree of freedom energies be further described on the potential for the entire quantum system energetic representations, I want "represents partially energetically" this quantum systems as well.

Since, in the following, the z-axis always defines the quantization direction of the quantum systems under consideration as well as those of the spin analyzers used, in order to simplify the notation, I would like to refer to the III. Refer to section introduced notation. The energetically higher state | + z> should therefore again be denoted by | 0> and the lower-lying state | -z> by | 1>.

One goes one step further, so the assumption is obvious that a coherently-coupled vacuum fluctuation in general, under the above conditions, which consist of Quantum systems comprising more than two sub-systems, is possible if the subsystems relevant to the state vector Ψ _- at least as inherent property can be assigned. This assumption can then be formulated in the form of the postulate (P1) as follows:
(P1): For a quantum system consisting of at least two subsystems, the energy μB _z can be exchanged between two subsystems in the spin degree of freedom by means of a coherently coupled vacuum fluctuation, if the energy of these subsystems of the quantum system in the spin degree of freedom both the energy μB _z and the energy -μB _z , about the energetic representations possible for the quantum system, can be assigned and to describe the state components in which these subsystems associated subspace, the base vector Ψ _{- is} required.

It should be noted that, via a coherently coupled vacuum fluctuation by definition, only those energetic representations of a quantum system are accessible by which different energies in the spin-degree of freedom are assigned to the respective subsystems. If energy representations are possible for a quantum system consisting of more than two subsystems, in which identical energies are assigned to the subsystems concerned, these energetic representations can not of course arise from coherently coupled vacuum fluctuations.

According to postulate (P1), a coherently coupled vacuum fluctuation is possible for a quantum system in state Ψ ₃₁ , but not for a quantum system in state | 0,1> or | 1,0>. Because: If, for example, a quantum system is in the state | 1,0>, then it can be written with respect to the base (B2) as: | 1,0> = 1.2 ^2.1 (Ψ _- Ψ + ²¹ ₊ ²¹⁾ (37)

In order to be able to describe the state | 1.0> with respect to the base (B2), the state vector Ψ _- is thus required, but the two subsystems in the spin-degree of freedom can not have both the energy μB _z and the energy -μB _{z '.} because the only possible energetic representation for a quantum system in the state | 1,0> is given by: "The spin degree of freedom of the subsystem S _L , as soon as the subsystem is in the magnetic field B _z , the energy E _spin = -μB _z and the spin degree of freedom of the subsystem S _R , as soon as the subsystem is in the magnetic field B _z , the energy E _spin = + μB _z ". Thus, according to postulate (P1), a coherently coupled vacuum fluctuation for a quantum system in the state | 1.0> can not occur.

The requirement that it be always possible for subsystems, which can exchange energy by means of a coherently coupled vacuum fluctuation, to assign both the energy μB _z and the energy μB _z to the energy representations possible for the quantum system thus ensures that that it is physically impossible to be able to selectively transfer any amount of energy from one subsystem to another subsystem by means of coherently coupled vacuum fluctuations.

By means of coherently coupled vacuum fluctuations, the energy μB _z could thus be exchanged in the spin-degree of freedom of two spin ½ particles, without these having interacted physically. Coherently coupled vacuum fluctuations could therefore open up the possibility of being able to detect the presence of entanglement even without a classical information channel in entangled quantum systems. I would like to show how this could be realized using the thought experiment described in Section VI. However, the question arises with what probability a coherently coupled vacuum fluctuation occurs? To answer this question, one must consider that a coherently coupled vacuum fluctuation can occur both in the transformation of an elementary quantum system into an energetically represented quantum system and in the transformation of an energetically represented quantum system into an elementary quantum system. It should also be noted that the state component Ψ _- can be assigned to the participating subsystems either as an inherent property or as a physically realized property. In general, therefore, this question can not be answered. However, there are clues for specific cases. On the basis of the considerations in Section VII, it is natural to assume that the transfer of an elementary quantum system into an energetically represented quantum system in which the quantum systems involved possess the state component Ψ _- as a physically realized property, will certainly lead to a coherently coupled vacuum fluctuation if this is possible after (P1). On the other hand, the above considerations suggest that if the state component Ψ _{- can} only be assigned as an inherent property to the participating subsystems, a coherently coupled vacuum fluctuation can occur only in the transformation of an energetically represented quantum system into an elementary quantum system since, for these quantum systems, the necessary boundary conditions for the occurrence of a coherently coupled vacuum fluctuation can probably only be created by means of the actually realized energetic representations of the quantum system. This assumption can easily be verified by means of the thought experiment described in Section VI. For this, only the length L _{M4 of} the magnetic field M _{4 has to be} varied. If L _{M4 is calculated} according to Eq. (41), a coherently coupled vacuum fluctuation in the thought experiment described in Section VI is likely to occur only if the assumption is incorrect. However, if the assumption is correct, a coherently coupled vacuum fluctuation should occur if: L _M1 = L _M4 and the thought experiment described there should then be able to be carried out successfully.

If it is possible, for example, that the longitudinal Stern-Gerlach effect can compete with the process of coherently coupled vacuum fluctuation, a coherently coupled vacuum fluctuation will no longer occur with certainty if this is possible after postulate (P1). The probability of occurrence of a coherently coupled vacuum fluctuation would then be <1.

• VI. Ein Gedankenexperiment zur wechselwirkungsfreien Informationsübertragung: Eine wesentliche Voraussetzung für das im Folgenden vorgeschlagene Gedankenexperiment ist, dass eines der beiden Teilsysteme S₁ oder S₂ (hier das Teilsystem S₁) so mit einem lokalen, statischen Magnetfeld wechselwirken kann (siehe ), dass für ein Quantensystem im Zustand Ψ₃₁, nachdem das Teilsystem S₁ das lokale Magnetfeld wieder verlassen hat, der Zustand erhalten bleibt. Im Wesentlichen entspricht der in dargestellte Aufbau dem im EPR-Bohm-Gedankenexperiment (siehe ) verwendeten Aufbau. Die Quelle Q_2/1 emittiert ein aus zwei Spin ½ Systemen zusammengesetztes Quantensystem im Zustand Ψ₃₁. Wobei sich die beiden Teilsysteme S₁ und S₂ wieder auf einer geraden Linie entlang der x-Achse in entgegengesetzter Richtung mit betragsmäßig derselben Geschwindigkeit u von der Quelle Q_2/1 entfernen. Entlang der x-Achse ist im Abstand A₁ rechts von der Quelle ein Magnete M₁ angeordnet. Dieser besitzt ein homogenes Magnetfeld B_z. Der Bereich mit homogenem Magnetfeld soll für den Magneten M₁ die Länge L_M1 haben. Die Übergangsbereiche vom feldfreien Raum in den Bereich mit dem homogenen Magnetfeld B_z sollen die Länge Δs_B haben. Mit Δt_B = Δs_B/u sei wieder die Flugzeit bezeichnet, die das Teilsysteme S₁ benötigt um die Strecke Δs_B zu durchfliegen. Die für die Zustandsanalyse der Teilsysteme S₁ und S₂ benötigten Zustandsanalysatoren sind mit SA₁ bzw. SA₂ bezeichnet. Die Quantisierungsrichtung der Zustandsanalysatoren SA₁ und SA₂ ist ebenfalls die z-Achse. Der Einfluss eines lokalen, konstanten Magnetfeldes auf den Zustand Ψ₃₁ wird in [37] für den hier relevanten Fall, wenn das lokale Magnetfeld nur mit einem Teilsystem (hier dem Teilsystem S₁) wechselwirkt, diskutiert. In welchen Zustand der Zustand Ψ₃₁ überführt wird, wenn das Teilsystem S₁ mit dem lokalen Magnetfeld B_z wechselwirkt, hängt von den jeweiligen Randbedingungen ab. Für eine realitätsnahe Beschreibung ist es erforderlich, die Geschwindigkeitsverteilung mit der die betrachteten Teilsysteme die Quelle Q_2/1 verlassen und die durch die räumliche Lokalisierung der Teilsysteme in der Quelle bedingte minimale Geschwindigkeitsunschärfe zu berücksichtigen. Im Folgenden soll daher immer die Randbedingung (R2) erfüllt sein: Δt_B ≤ Δt_max = h/(4πΔE_z/2) (R2.1) (u₀ – Δu_Q/2) < u < (u₀ + Δu_Q/2), mit Δu_Q > 0 und Δu_Q < 10^–3u₀ (R2.2) Δs_B » A₁10^–3 (R2.3) |Δu_MH| < 10^–2Δu_min < Δu_Q (R2.4) f_lamΔt_MB1 ≤ 2, mit f_lam = ΔE_z/h und Δt_MB1 = (L_M1 + ΔS_B)/u, (R2.5) wobei mit u₀ die mittlere Geschwindigkeit der Teilsysteme, mit Δu_Q das Intervall in dem die Geschwindigkeit u variiert, mit Δu_MH die, mit dem Eintritt des Teilsystems S₁ in das Magnetfeld B_z, aufgrund des longitudinalen Stern-Gerlach-Effektes bedingte Geschwindigkeitsänderung, mit Δu_min die über die Unschärferelation für Ort und Impuls bedingte minimale Geschwindigkeitsunschärfe für die Teilsysteme S₁ und S₂, mit f_lam die Lamorfrequenz und mit Δt_MB1 die effektive Zeit die das Teilsystem S₁ benötigt, um das Magnetfeld B_z zu durchqueren, bezeichnet sei.

Since the following considerations can also be applied to the case where the probability of occurrence of coherently coupled vacuum fluctuation is <1, the argument can be made particularly simple by assuming that a coherently coupled vacuum fluctuation under the boundary conditions described above In the first thought experiment described in Section VI, I would like to assume that a coherently coupled vacuum fluctuation will always occur with certainty if this is possible according to (P1).

• VI. A thought experiment for interaction-free information transfer: An essential prerequisite for the thought experiment proposed below is that one of the two subsystems S ₁ or S ₂ (here the subsystem S ₁ ) can interact with a local, static magnetic field (see ) that for a quantum system in the state Ψ ₃₁ , after the subsystem S _{1 has left} the local magnetic field again, the state is maintained. Essentially, the in shown structure in the EPR Bohm thought experiment (see ) used construction. The source Q _2/1 emits a composite of two spin ½ systems quantum system in the state Ψ _31st Where the two subsystems S ₁ and S ₂ again on a straight line along the x-axis in the opposite direction with magnitude the same speed u from the source Q _2/1 remove. Along the x-axis, a magnet M _{1 is} arranged at the distance A _{1 to the} right of the source. This has a homogeneous magnetic field B _z . The homogeneous magnetic field should have the length L _M1 for the magnet M ₁ . The transition areas from the field-free space to the area with the homogeneous magnetic field B _z should have the length Δs _B. With Δt _B = Δs _B / u again the flight time is called, which requires the subsystems S ₁ to fly through the distance Δs _B. The state analyzers required for the state analysis of the subsystems S ₁ and S ₂ are designated SA ₁ or SA ₂ . The quantization direction of the state analyzers SA ₁ and SA ₂ is also the z-axis. The influence of a local, constant magnetic field on the state Ψ ₃₁ is discussed in [37] for the case relevant here, when the local magnetic field interacts with only one subsystem (here the subsystem S ₁ ). Is in which state of the state Ψ transferred ₃₁ when the subsystem S ₁ with the local magnetic field B _z interacts depends on the respective boundary conditions. For a realistic description, it is necessary to leave the velocity distribution with the considered subsystems the source Q _2/1 and to take into account the minimum velocity uncertainty caused by the spatial localization of the subsystems in the source. In the following, therefore, always the boundary condition (R2) should be fulfilled: Δt _B ≤ Δt _max = h / (4πΔE _z / 2) (R2.1) (u ₀ -Δu _Q / 2) <u <(u ₀ + Δu _Q / 2), with Δu _Q > 0 and Δu _Q <10 ^-3 u ₀ (R2.2) Δs _B »A ₁ 10 ^-3 (R2.3) | Δu _MH | <10 ^-2 Δu _min <Δu _Q (R2.4) f _lam Δt _MB1 ≤ 2, with f _lam = ΔE _z / h and Δt _MB1 = (L _M1 + ΔS _B ) / u, (R2.5) where with u ₀ the average velocity of the subsystems, with Δu _Q the interval in which the velocity u varies, with Δu _MH the, with the entry of the subsystem S ₁ in the magnetic field B _z, due to the longitudinal Stern-Gerlach effect speed change , where Δu _{min is} the minimum velocity uncertainty for the subsystems S ₁ and S _{2 due} to the uncertainty relation for location and momentum, where f _{lam is} the Lamor frequency and Δt _{MB1 is} the effective time required by the subsystem S ₁ to traverse the magnetic field B _z , is designated.

As one easily checks, then the state Ψ ₃₁ by the influence of the magnet M ₁ on the subsystem S ₁ in a good approximation in the pure state Ψ ₃₈ = 1/2 ^1/2 e ^{-iθ / 2} (| 1.0> ^{-e -iθ} | 0.1>), (38) With θ = 2πf _lam Δt _MB1 transferred. If one chooses f _lam and Δt _MB1 such that f _lam Δt _MB1 = 1, (39) so we get back to the global phase, the state Ψ ₃₁ . The phase e ^iθ results because the magnetic moment μ was generally assumed to be anti-parallel to the spin (see above). If the magnetic moment is aligned parallel to the spin, the phase e ^{-iθ results} . The condition (R2.1) corresponds to the condition (R1.1). The condition (R2.2) describes the velocity distribution of the subsystems caused by the source Q and together with condition (R2.3) ensures that Δt _{B is} substantially greater than the differences in the entry times of the subsystems S ₁ due to the velocity distribution in the transition region of the magnet M ₁ . Since it is ensured by conditions (R2.4) and (R2.5) that the spatial part of the state remains almost unchanged, I would like to continue suppressing it. The deviations of the amplitudes of the components | 0,1> and | 1,0> are smaller than ± 10 ^-4 under these boundary conditions. The deviations for the phase angle θ are less than ± 2.1 mrad (± 0.75 °).

Of course, one can ask oneself whether it is even possible to fulfill the boundary condition (R2)? I would like to illustrate this with a simple example:
Assuming that the entangled system is made up of two ⁴⁰ Ca ⁺ ions and spatially localized to approximately 1 μm in the source Q during the preparation process, the uncertainty relation for location and momentum gives the velocity of the two Subsystems a minimum velocity uncertainty Δu _min = 7.9 · 10 ^-4 m / s. The transition region of the magnet M ₁ from field-free space in the area with a homogeneous magnetic field B _z should here have a length of Δs _B = 0.2 mm. If one further chooses for u ₀ = 10.429.8 m / s and over condition (R2.1): 3 · Δt _B = Δt _max , the result for Δt is _B = 19 ns and ΔE _z = 1.8 · 10 ^-27 (1.27 x 10 ^-8 eV). For the local magnetic field B _z , the value B _z = 1 G (10 ^-4 T) is then obtained. For the velocity change caused by the longitudinal Stern-Gerlach effect, the value Δu _MH = ± 1.3 · 10 ⁴ m / s is obtained. If one chooses for L _MB1 = (L _M1 + Δs _B ) = 3.57 mm, the result for the phase e ^iθ = 1 is that θ then just assumes the value 2π. The source Q could be realized by the linear ion trap described in [17]. This would require the ion trap (see ) superimposed magnetic field are switched off, after the two ions were prepared in the state Ψ ₂₅ . If the two electrodes arranged at the ends of the ion trap are provided with a bore (along the axis of the ion trap), the ions can be released from the ion trap at a defined time at a defined speed. For this purpose, it is only necessary to switch off the positive voltage applied to the two electrodes with a defined time ramp. Since voltages can now be switched with high precision, it should be possible to fulfill condition (R2.2) as well. Also, (R2.3) can be satisfied, since the ion trap used in [17] has only a length of about 3 cm. Since in this example the two subsystems carry S ₁ and S _2, a charge must, if the subsystem S ₁ with the local magnetic field B _z of the magnet M ₁ interacts, the influence of the Lorentz force are taken into account the state of the system. Although the Lorenz force only interacts with the overall system via the charge of the subsystem S ₁ , it can, due to the longitudinal Stern-Gerlach effect, depending on whether the subsystem has the energy ΔE _z / 2 in the spin-degree of freedom with entry into the magnetic field B _z picks up or give, lead to different distances covered by the subsystem S ₁ in the magnetic field B _z . However, as is easily checked, the difference between them under the boundary conditions given here in the paths is less than 3 · 10 ^-19 m and therefore negligible. The boundary condition (R2) can therefore, as this example shows, be fulfilled. As state analyzers, however, no Stern-Gerlach magnets can be used in this case. The reason for this is that the Lorenz force makes a separation of the spin states impossible. However, it should, as is shown in [38], also for charged subsystems by means of a longitudinal magnetic field which has a gradient, be possible to detect the spin state of the subsystems S ₁ and _S2.

Below I would like to see the in structure described as a modified EPR Bohm structure (EPRBM _M1 structure), when the boundary condition (R2) is met. The index "M1" is intended to indicate thereby that the subsystem S ₁ passes through the right ordered by the source Q M magnets. ₁

schematically shows the basic structure for a thought experiment for interaction-free information transmission. In the thought experiment two modified EPR-Bohm abutments EPRBM _M1 and EPRBM _M4 are used, which should be arranged at a distance d from each other. These differ only in that L _M4 ≥ L _M1 should be. Otherwise the two modified EPR-Bohm abutments EPRBM _M1 and EPRM _{M4 are} identical. The magnets M ₁ and M ₄ arranged to the right of the sources Q _2/1 and Q _3/4 are arranged at a distance A ₁ = A ₄ from the respective sources. The magnets M ₁ and M ₄ should have the same magnetic field B _z and the same quantization axis z. The state analyzers SA ₁ , SA ₂ , SA ₃ and SA ₄ also all have the z-direction as the quantization axis. The source Q _{2,1 of} the construction EPRBM _M1 the subsystem S ₂ and S _{1 should} emit in the state Ψ ₃₁ and the source Q _{3/4 of} the structure EPRBM _M4 should also emit the subsystems S ₃ and S ₄ in the state Ψ ₃₁ . All subsystems leave the respective sources with the same speed u. If the two sources Q _2,1 and Q _3,4 are operated synchronously in time, the subsystems S ₁ and S ₄ simultaneously reach the magnets M ₁ and M _4, respectively. In order to be able to exclude with certainty that a coherently coupled vacuum fluctuation can occur when the subsystems enter the individual state analyzers, the distances of the state analyzers to the corresponding sources should be selected appropriately (see also the remarks on Eq. (41)). In the following, I assume that the two sources simultaneously emit a respective system consisting of the two subsystems S ₁ and S ₂ or S ₃ and S ₄ in state Ψ ₃₁ at previously defined times.

Following the considerations in Section V, there are two ways in which the thought experiment could be realized:
Variant 1: The process of the coherently coupled vacuum fluctuation can occur both at the entry and at the exit of the subsystems S ₁ and S ₄ from the magnetic fields M ₁ and M ₄ . In this case, L _M4 > L _M1 (taking Eq. (41)) must be chosen so that the process of coherently coupled vacuum fluctuation can occur only when the subsystems enter the magnetic fields.
Or:
Variant 2: The process of coherently coupled vacuum fluctuation can only occur when the subsystems S ₁ and S ₄ emerge from the magnetic fields M ₁ and M ₄ . In this case, L _M4 = L _M1 would have to be selected.

Since it is not possible to decide for which variant the thought experiment can be carried out successfully, I would like to briefly describe both variants. First, I would like to explain the thought experiment in case it can be realized according to variant 1:
In the structure EPRBM _M1 and EPRBM _M4 , the lengths L _M1 and L _M4 are selected such that the following applies: f _lam Δt _MB1 = 1 and f _lam Δt _MB4 = 2 (40) With: Δt _MB1 = (L _M1 + Δs _B ) / u and Δt _MB4 = (L _M4 + Δs _B ) / u.

As a result, it is ensured in compliance with the boundary condition (R2) that the phase ^{e.sub.i.sub.θ} caused by the longitudinal Stern-Gerlach effect in Eq. (38), both for the existing of the subsystems S ₁ and S ₂ quantum system in the state Ψ ₃₈ , as well as for the existing of the subsystems S ₃ and S ₄ quantum system in the state Ψ _38, the value e ^iθ = 1 assumes. Next should apply: (L _M4 - L _M1 ) ≥ Δt _vir u (41)

This ensures that with the exit of the subsystems S ₁ and S ₄ from the respective magnetic fields of the longitudinal Stern-Gerlach effect occurs and thus can be ruled out with certainty that the subsystems can interact via a coherently coupled vacuum fluctuation.

I would like to call the operator of the EPRBM _M1 body "Alice" and the operator of the EPRBM _{M4 body} "Bob". Furthermore, I would like to assume that between Alice and Bob a transmission protocol has been arranged in such a way that Alice, at predetermined time intervals, also sources Q _2/1 , asynchronous from the source Q ₃ , deviating from the times previously agreed for synchronous operation. ₄ of Bob if you want this. The same shall apply to Bob, whereby the agreed time intervals should not overlap and the sources Q _2/1 and Q _3/4 for the asynchronous operation should emit the respective subsystems at least by the interval against each other with a time delay. According to this, there should be no physical interaction between the EPRBM _M1 and EPRBM _M4 bodies. There should be no classical exchange of information between Alice and Bob anymore.

Since the structures EPRBM _M1 and EPRBM _M4 can be considered as physically independent systems thereafter, the quantum physical state of the total system consisting of the four subsystems S ₁ , S ₂ , S ₃ and S ₄ when the source Q _{2/1 is} the subsystems S ₁ and S ₂ emitted in the state Ψ ₃₁ and the source Q _3/4, the subsystems S ₃ and S ₄ in the state Ψ ₃₁ emitted by Ψ _- ^43/21 with Ψ _- ^43/21 = Ψ _- ⁴³ × Ψ _- ²¹ (42) With: Ψ _- ⁴³ = 1/2 ^1/2 (| 1> ₄ | 0> ₃ - | 0> ₄ | 1> ₃ ) Ψ _- ²¹ = 1/2 ^1/2 (| 1> ₂ | 0> ₁ - | 0> ₂ | 1> ₁ ) given. Where the indices "1", "2", "3" and "4" denote the respective subsystems. For the spin analyzers SA ₁ , SA ₂ , SA ₃ and SA ₄ used for the state analysis, therefore, regardless of whether the sources Q _2/1 and Q _3/4 are operated synchronously or asynchronously, the values shown in Table 1 given measured values.

The measurement result 0 (1) means that the respective subsystem with the measurement is detected in the energetically higher (energetically lower) state | 0> (| 1>) (if the Stern-Gerlach magnets described above are used as the state analyzers, is deflected in the direction + z (-z)). The readings available to Alice via the spin analyzers SA ₁ and SA ₂ should therefore always be strictly anti-correlated in the ideal case. Also, the readings available to Bob, via the spin analyzers SA ₃ and SA ₄ , should ideally be strictly anti-correlated in the ideal case.

However, which measured values would one expect if postulate (P1) is true and as a consequence of this coherent coupled vacuum fluctuations can occur between subsystems S ₁ and S ₄ ? According to postulate (P1), in order for a coherently coupled vacuum fluctuation to occur between the subsystems S ₁ and S ₄ , both subsystems must be assigned both the energy μB _z and the energy μB _z in the spin degree of freedom via the energetic representations possible for the quantum system can be. This is obviously possible according to Table I. Furthermore, these subsystems must be able to assign the state vector Ψ _- at least as an inherent property in the subspace allocated to these subsystems (the state vector Ψ _- must be required to determine the state component of the subsystems S ₁ and S ₄ , on the subspace allocated to these subsystems, to be able to describe the basis (B2)). Now you can rewrite the state Ψ _- ^43/21 Ψ _- ^43/21 = 1/4 ^1/2 {Ψ ₊ ^41/32 - Φ ₊ ^41/32 + Ψ _- ^41/32 + Φ _- ^41/32 } (43) With: Ψ ₊ ^41/32 = 1/2 ^1/2 {| 1> ₄ Ψ ₊ ³² | 0> ₁ + | 0> ₄ Ψ ³² | 1> ₁ } Φ ₊ ^41/32 = 1/2 ^1/2 {| 0> ₄ Φ ₊ ³² | 0> ₁ + |> ₄ Φ ₊ ³² | 1> 1} Φ _- ^41/32 = 1/2 ^1/2 {| 0> ₄ Φ _- ³² | 0> ₁ - | 1> ₄ Φ _- ³² | 1> ₁ } Ψ _- ^41/32 = 1/2 ^1/2 {| 1> ₄ Ψ _- ³² | 0> ₁ - | 0> ₄ Ψ _- ³² | 1> ₁ } and: Ψ ₊ ³² = 1/2 ^1/2 (| 1> ₃ | 0> ₂ + | 0> ₃ | 1> ₂ ) Φ ₊ ³² = 1/2 ^1/2 (| 0> ₃ | 0> ₂ + | 1> ₃ | 1> ₂ ) Ψ _- ³² = 1/2 ^1/2 (| 1> ₃ | 0> ₂ - | 0> ₃ | 1> ₂ ) Φ _- ³² = 1/2 ^1/2 (| 0> ₃ | 0> ₂ - | 1> ₃ | 1> ₂ ).

According to Eq. (43) this is obviously possible. Thus, according to postulate (P1), a coherently coupled vacuum fluctuation can occur between the subsystems S ₁ and S ₄ . The same naturally also applies to the subsystems S ₂ and S ₃ .

I would first like to look at the case when sources Q _2/1 and Q _{3/4 are} operated synchronously. An overview of the cases of state development possible for the quantum system is obtained by considering the formally possible cases for the settings of the spins of the individual subsystems. In the two then possible, in principle indistinguishable cases for the state development of the state of the whole system are shown. The indices "1", "2", "3" and "4" again denote the respective ones Subsystems. "+" Means that a parallel to the magnetic field B _z aligned spin state component is assigned to the corresponding subsystem and "-" means that the corresponding subsystem is assigned an antiparallel to the magnetic field B _z aligned spin state component. With M ₁ and M ₄ , the magnetic fields are again referred to, with which the subsystems S ₁ and S ₄ interact.

In the case (I :), no coherently coupled vacuum fluctuation between the subsystems S ₁ and S ₄ can occur since the spin state components associated with the subsystems match. For this case, the state vector Ψ _- ^{43/21 is} retained for the entire system. In the case (II), however, a coherently coupled vacuum fluctuation occurs and reverses the spin state components associated with the subsystems S ₁ and S ₄ . At least so much is established about the state vector to be assigned to the overall state in this case, that this subspace on the subspaces defined by the subsystems S ₁ and S ₂ or S ₃ and S _{4 is} respectively represented by the fractions | 0.0> and | 1 , 1> must be writable. Cases (I.) and (II.) Thus occur with a probability of 50% each. In the event that the coherently coupled vacuum fluctuation does not occur with certainty for the case (II.), Then the probability for this case would decrease accordingly.

With this information, it is now possible to determine the state of the overall system, which results after the subsystems S ₁ and S _{4 have} interacted with the magnets M ₁ and M ₄ and have left the corresponding magnetic fields again.

Within quantum physics, any lossless state transformation (one that gets the norm) must be described by a unitary operator. Of course, this also applies to the preparation process considered here by means of coherently coupled vacuum fluctuations. The state Ψ _- ^43/21 can be rewritten Ψ _- ^43/21 = 1/2 ^1/2 {1/2 ^1/2 Ψ ₊ ^41/32 - Φ ₊ ^41/32 ) - 1/2 ^1/2 (Ψ _- ^41/32 - Φ _- ^41/32 )} = 1/2 ^1/2 (| A> - | B>) (44) With: | A> = 1/2 ^1/2 (Ψ ₊ ^41/32 - Φ ^41/32 ) and | B> = 1/2 ^1/2 (Ψ _- ^41/32 - Φ _- ^41/32 ), where: <A | B> = 0. (45)

so erhält man: U_KV(Ψ_– ^43/21) = |A> = 1/2^1/2(Ψ₊ ^41/32 – Φ₊ ^41/32) (47) The state Ψ _- ^43/21 can thus also be formally understood as an element of a two-dimensional vector space ^spanned by the orthogonal state vectors | A> and | B>. If we define the unitary operator U _KV according to this vector space

so you get: U _KV (Ψ _- ^43/21 ) = | A> = 1/2 ^1/2 (Ψ ₊ ^41/32 - Φ ₊ ^41/32 ) (47)

The state | A> can be rewritten as: | A> = 1/2 ^1/2 (Ψ ₊ ^41/32 - Φ ₊ ^41/32 ) = 1/2 ^1/2 (Ψ _- ^43/21 - Φ _- ^43/21 ) (48) With: Φ _- ^43/21 = Φ _- ⁴³ × Φ _- ²¹ and: Φ _- ⁴³ = 1/2 ^1/2 (| 0> ₄ | 0> ₃ - | 1> ₄ | 1> ₃ ) Φ _- ²¹ = 1/2 ^1/2 (| 0> ₂ | 0> ₁ - | 1> ₂ | 1> ₁ )

This results in Eq. (47): Ψ ₄₉ = U _KV (Ψ _- ^43/21 ) = 1/2 ^1/2 (Ψ _- ^43/21 - Φ _- ^43/21 ). (49)

The state resulting from the process of coherently coupled vacuum fluctuation must consist of two states according to the above considerations. The state vector Ψ _- ^43/21 and a state part of the sub-spaces defined by subsystems S ₁ and S ₂ or S ₃ and S ₄ can be described by the components | 0.0> and | 1.1>. For the absolute squares of the respective state shares, it was required that they have the value 1/2. State Ψ ₄₉ fulfills these conditions. The state transformation of the four subsystems S ₁ , S ₂ , S ₃ and S ₄ in the state Ψ ^43/21 , which is conditioned by the process of the coherently coupled vacuum fluctuation, can thus be described by the unitary operator U _KV . In the calculation base for state Ψ ₄₉ you get: Ψ ₄₉ = 1/8 ^1/2 {| 1> ₄ | 0> ₃ | 1> ₂ | 0> ₁ - | 1> ₄ | 0> ₃ | 0> ₂ | 1> ₁ - | 0> ₄ | 1 > ₃ | 1> ₂ | 0> ₁ + | 0> ₄ | 1> ₃ | 0> ₂ | 1> ₁ - | 0> ₄ | 0> ₃ | 0> ₂ | 0> ₁ + | 0> ₄ | 0> ₃ | 1> ₂ | 1> ₁ + | 1> ₄ | 1> ₃ | 0> ₂ | 0> ₁ - | 1> ₄ | 1> ₃ | 1> ₂ | 1> ₁ } (50)

From Eq. (50) one can now read directly the possible measured values and the probabilities with which they occur. Alice and Bob now no longer receive exclusively strictly anti-correlated measurements on the state analyzers that are accessible to them, but also correlated measured values in 50% of the cases.

Only when Alice or Bob decide in the previously agreed time intervals to operate the respective source asynchronously, both again receive strictly anti-correlated measured values, since then with the entry of the subsystems S ₁ and S ₄ in the respective magnetic fields not coherently coupled Vacuum fluctuation can occur.

In this case, the measured values given in Table I are again obtained. Also in the case that coherently coupled vacuum fluctuations do not occur with certainty, if these are possible after postulate (P1), one obtains qualitatively the same result. Corresponding to the probability of the occurrence of a coherently coupled vacuum fluctuation, only the probability of correlated measured values on the spin analyzers accessible to Alice and Bob is reduced.

If the thought experiment according to variant 2 is realized, it is only to be noted that the condition in Eq. (40) passes into the condition f _lam Δ _MB1 = f _lam Δt _MB4 = 1 and Eq. (41) of course does not have to be fulfilled. In contrast to variant 1, if this is possible according to (P1), a coherently coupled vacuum fluctuation does not occur with the entry of the subsystems into the magnetic fields, but only with the emergence of the subsystems from the magnetic fields. Since there are no further differences between the two variants, both variants lead to the same result. If the structures EPRBM _M1 and _M4 EPRBM operated synchronously, so again the state of the quantum system can Ψ after the subsystems S have the magnetic fields M ₁ and M ₄ leave ₁ and S ₄ are assigned to _{the 49th} If the assemblies EPRBM _M1 and EPRBM _M4 are operated asynchronously, the state of the quantum system can again be assigned the state Ψ _- ^43/21 after the subsystems S ₁ and S ₄ have again left the magnetic fields M ₁ and M ₄ .

If the thought experiment can be realized and it turns out that the readings available for Alice and Bob depend on whether the sources Q _2/1 and Q _3/4 are operated synchronously or asynchronously, this would of course be a strong indication that Postulate (P1) is true and coherently coupled vacuum fluctuations can occur under the appropriate boundary conditions. Since the ideas underlying the postulate (P1) are based on the interpretation approach proposed here, it is also justified to assume that it adequately describes the essential aspects of quantum physics and provides a sound starting point for an adequate interpretation of quantum physics and quantum physics Idea forms.

• I.) Wenn die für Alice und Bob zugänglichen Messwerte davon abhängen, ob die Quellen Q_2/1 und Q_3/4 synchron oder asynchron betrieben werden, kann die physikalische Welt nicht kausal abgeschlossen sein, da hier angenommen wurde, dass es zwischen den modifizierten EPR-Bohm-Aufbauten EPRBM_M1 und EPRBM_M4, keinerlei physikalische Wechselwirkungen geben soll, sobald diese in Betrieb genommen werden. Vermittelt über den Prozess der kohärent gekoppelten Vakuumfluktuationen könnten physikalische Ereignisse somit auch dann kausal zusammenhängen, wenn es keinerlei Möglichkeiten für eine physikalische Wechselwirkung zwischen den betrachteten Quantensystemen gibt.
• II.) Wenn die für Alice und Bob zugänglichen Messwerte auch dann noch davon abhängen, ob die Quellen Q_2/1 und Q_3/4 synchron oder asynchron betrieben werden, wenn der Abstand d zwischen den modifizierten EPR-Bohm-Aufbauten EPRBM_M1 und EPRBM_M4 beliebig groß gewählt werden kann, so wäre dieses ein Indiz dafür, dass die Energie μB_z augenblicklich über beliebige Entfernungen ausgetauscht werden kann. Sofern die für Alice und Bob zugänglichen Messwerte nur für d < ch/(4πΔE_z/2), davon abhängen, ob die Quellen Q_2/1 und Q_3/4 synchron oder asynchron betrieben werden, würde das bedeuten, dass auch für kohärent gekoppelte Vakuumfluktuationen eine, wenn auch wechselwirkungsfreie, endliche Übertragungsgeschwindigkeit für die Energie μB_z besteht.

For the physical world view, the thought experiment described above has a fundamental meaning in two respects:

• I.) If the readings accessible to Alice and Bob depend on whether sources Q _2/1 and Q _3/4 are operated synchronously or asynchronously, the physical world can not be causally closed, since it was assumed to be between the two Modified EPR-Bohm abutments EPRBM _M1 and EPRBM _{M4 should} not give any physical interaction as soon as they are put into operation. Thus, given the process of coherently coupled vacuum fluctuations, physical events could be causally related even if there were no possibilities for a physical interaction between the considered quantum systems.
• II.) If the readings available for Alice and Bob still depend on whether the sources Q _2/1 and Q _3/4 are operated synchronously or asynchronously, if the distance d between the modified EPR-Bohm setups EPRBM _M1 and EPRBM _M4 can be chosen arbitrarily large, this would be an indication that the energy μB _z can be exchanged immediately over any distance. If the readings available to Alice and Bob depend only on d ch / (4πΔE _z / 2), depending on whether sources Q _2/1 and Q _3/4 are operated synchronously or asynchronously, this would mean that they are also coherent coupled vacuum fluctuations an, albeit interaction-free, finite transmission speed for the energy μB _z exists.

For a better understanding of the thought experiment described above, I would like to briefly summarize the basic idea (see ):
Let's assume that two friends, let's call you Alice and Bob, could do the following experiment: Alice has a construction with a source Q _1/2 which at a freely selectable time has two spin ½ systems S ₁ and S ₂ in the singleton Condition Ψ _- can emit. The systems S ₁ and S ₂ are to meet the condition analysis according to a defined route on the spin analyzers SA ₁ and SA ₂ . Between the source Q _1/2 and the spin analyzer SA ₁ should be located locally a unit E ₁ , which must pass through the system S ₁ , before it reaches the spin analyzer SA ₁ . The structure of Bob has a source Q _3/4 at a freely selectable time also two spin ½ systems S ₃ and S ₄ in the singlet state Ψ _- can emit. The systems S ₃ and S ₄ are to meet the state analysis for a defined route on the spin analyzers SA ₃ and SA ₄ . Between the source Q _3/4 and the spin analyzer SA ₄ should be located locally a unit E ₄ , which must pass through the system S ₄ , before it reaches the spin analyzer SA ₄ . The spin analyzers should be arranged so that the systems S ₁ , S ₂ , S ₃ and S ₄ reach them only after the systems S ₁ and S ₄ the Leave units E ₁ and E ₄ again. All spin analyzers should have the same quantization axis (here the z-axis). The abutments of Alice and Bob should be arranged at the distance d from each other. With the commissioning of the superstructures, there should no longer be a physical interaction between the superstructures. Also, no information should be exchanged between Alice and Bob on classical channels. The construction of Alice and the construction of Bob should be further realized so that in the event that the sources Q _1/2 and Q _{3/4 are} operated synchronously (both sources simultaneously emit a quantum system in the singlet state Ψ _- ), the systems S ₁ and S ₄ also simultaneously reach the units E ₁ and E ₄ . With the commissioning of the experiment Alice and Bob are no longer able to influence the units E ₁ and E ₄ . The units E ₁ and E ₄ should be able to influence the state of the overall system in such a way that the following measured values result: If the sources Q _1/2 and Q _{3/4 are} operated synchronously, then the measured values accessible to Alice (measured values of the spin Analyzers SA ₁ and SA ₂ ) show no correlations. The same applies to the measured values of the spin analyzers SA ₃ and SA ₄ accessible to Bob. If the sources are operated asynchronously (the sources Q _1/2 and Q _3/4 emit the quantum systems offset by a defined time interval (see also Equation 32)), the measured values accessible to Alice are always strictly anti -correlated. The same should apply to the readings available to Bob. It is obvious that this thought experiment would open up the possibility of - in principle - instantaneous transfer of information. However, Alice and Bob would have to agree on a corresponding transmission protocol for this:
In order to be able to operate the sources synchronously, Alice and Bob always agree on the full minutes as times. To be able to operate the sources asynchronously, Alice and Bob agree on time intervals in which the apparatuses can also be operated asynchronously (for example, delayed by one second to the full minute). For example, between 8:00 am and 9:00 am (10:00 am and 11:00 am, 12:00 am, and 1:00 pm, etc.) Alice can freely decide whether her source Q _1/2 emits the systems exactly at the full minute or only delays them by one second. The same should apply to Bob. Bob will then be free to decide between 9am and 10am (11am and 12pm, 1pm and 2pm, etc.) if his Q _3/4 source emits the systems at exactly the full minute or decelerates for a second. Only in these defined time intervals, Alice and Bob are free to decide when their sources should emit the systems. Outside these time intervals, the sources must each emit the systems at full minute.

Are the units E ₁ and E ₄ by the in the realized magnetic fields M ₁ and M ₄ realized, so corresponds to in shown construction in the shown construction. The state dynamics caused by the units E ₁ and E ₄ can thus be described for the asynchronous operation simply by the identity operator (nothing happens). As measured values, Alice and Bob would always receive strictly anti-correlated values from the spin analyzers available to them. For synchronous operation, the state dynamics produced by the units E ₁ and E _{4 can} then be described by the operator U _KV . On the readings of the spin analyzers available to you then Alice and Bob would no longer find any correlations.

To be able to exchange information, Alice and Bob now only have to decide to which operating state which information should be assigned. For example, synchronous operation could be assigned a logical "one" and asynchronous operation a logical "zero". Alice and Bob could thus exchange information in a way that is in principle impossible by means of a classical information channel. A classic information channel always needs a sender and receiver. In order to transmit information from the transmitter to the receiver, the transmitter must be able to transmit energy to the receiver. Whether the receiver has received the information, the sender can not know for sure. However, if Alice or Bob, in their assigned time intervals, opt for synchronous operation, both would always know that the superstructures are operated synchronously. Since then the two structures would prepare an entangled quantum system, this knowledge could be present with almost no delay. It would thus be conceivable that Alice and Bob could also transmit information in a light-fast manner in this way.

Again and again, as an argument that such a thought experiment can not be realized in principle, it is stated that it follows from the special theory of relativity that signals can only be transmitted at maximum speed of light. This objection is then often supplemented by the observation that the assumption of an ultra-fast information transfer results in compulsory temporal paradoxes, which would then lead to inconsistencies within the theory used. Both objections, however, are not tenable.

The basic problem with this argument is that it usually implies that a sender and a receiver are needed to exchange information. This idea, shaped by classical physics, seems to have attained the status of an irrefutable fact, which therefore need not be further questioned. Obviously, this idea is also maintained in the context of quantum physics (his example, [12], [28]). In order to be able to prove that this idea can also be maintained in the context of quantum physics, the EPR-Bohm thought experiment is usually considered. But what can be concluded from this thought experiment? The essential point in this context is the following: By no measurement of any kind on a subsystem, the statistical distribution of the possible measured values on the other subsystems can be changed. Together with the fact that individual measurements can not be predicted in principle, if several measurements are possible (without this possibility is an information transfer not possible), then it can be concluded for the EPR Bohm thought experiment that information from a subsystem to a other subsystem can only be transmitted if in addition a classic information channel is available. However, the EPR-Bohm thought experiment can not be cited as proof that, in principle, a classical channel is required for information transmission in the context of quantum physics, because for this it would have to be shown that quantum physics can not provide any other means of transmitting information , One would therefore have to be able to prove the following statement: Because of the quantum physics possible instantaneous effect between quantum systems (which obviously is not related to an exchange of energy between the involved quantum systems) and to which the EPR in [9] pointed, there is an exchange of information (without an additional classical channel) in principle impossible. Only if this assertion is correct, it can be concluded that the classical notion must be valid. For an exchange of information, a sender and a receiver would then always be required and the sender would necessarily have to transmit energy to the receiver for the purpose of exchanging information. So the question is: is there any proof of this claim? The answer is no. So far, there is no evidence for this assertion. Also, I find it very unlikely that such a proof in the context of quantum physics is generally possible. In order to be able to clarify whether this claim is correct, there remains only the possibility of finding a counter-example for this. If this succeeds, the claim can not be correct. But how do you find a corresponding counter-example? For this it is necessary to consider which physical conception underlies this assertion. In order to understand what this concept is, it is only necessary to consider which physical properties are usually assigned to state transformations that describe preparation processes that lead to entangled quantum systems. For this purpose, the techniques described in Section III for state preparation of ⁴⁰ Ca ⁺ ions stored in a linear ion trap. If two ⁴⁰ Ca ⁺ ions are stored in the linear ion trap and both ions are present in the energetic ground state, the state describing the entire system is given by the product state | 1,1>. In order to convert the product state | 1,1 into an entangled state, a non-local unitary operator is required (see the comments at the beginning of Section VII). For example, the state | 1,1> may be formalized by means of the non-local unitary operator U _B2B1 ^-1 (see Equation 14) according to U _B2B1 ^-1 | 1,1> = Ψ _- = 1/2 ^1/2 (| 1 , 0> - | 0.1>) in the maximum entangled state Ψ _{- be} transferred. It is clear that the operator U _B2B1 ^{-1 describes} a formally permissible state transformation in the context of quantum physics. But what physical meaning can be attributed to the operator U _B2B1 ^-1 ? In order to be able to physically realize the operator U _B2B1 ^-1 by means of the preparation method described in Section III, it is imperative that there is an interaction between the two ions (there the Coulomb repulsion). The operator U _B2B1 ^-1 is therefore usually regarded as an interaction operator. Experiments of this kind thus seem to suggest the following assumption: Any physically feasible non-local unitary operator that transforms a product state into an entangled state must be an interaction operator. The crucial point is that in quantum physics it is neither shown that every physically realizable non-local operator has to be an interaction operator, nor can it be shown that there are physically feasible non-local operators in quantum physics that do not have any Interaction operators are. Notwithstanding this fact, however, quantum physics seems to have established the notion that any physically feasible non-local unitary operator that transforms a product state into an entangled state must be an interaction operator. The reason for this can probably be seen only in our, characterized by classical physics, intuition. If measurements are carried out on isolated systems and correlations occur between the measured values on the individual systems, there must have been an interaction between the systems in the past if the observed correlations have a causal cause. The interaction then determines the possible correlations. However, the EPR-Bohm thought experiment shows that this classical concept can not be applicable in the context of quantum physics, since in this case correlations between the two subsystems are possible, which are not classically explainable via interactions during preparation (see the remarks on Beginning of section VII). But this leads directly to the question of what role interactions play in the preparation of entangled systems? Considering the preparation procedure described in Section III to prepare two ⁴⁰ Ca ⁺ ions starting from the product state | 1,1> in the maximum entangled state Ψ _- , one sees that the Coulomb repulsion between the two ions is only ensures that it is in principle impossible to predict which energy in the state of the individual ions Ψ _- can be assigned. Only if one assumes that this goal can be realized exclusively by means of an interaction, can any physical-transferable non-local unitary operator, describing a state transformation, be understood as an interaction operator. An interaction-free transmission of information would then be possible, as in the EPR-Bohm thought experiment, only by means of an additional classical channel, since experiments such as the thought experiment described above would then be impossible in principle. The idea that in quantum physics, in principle, an additional classical information channel is needed to exchange information would then be correct. However, once one assumes that there may be physically feasible non-local unitary operators that can not be understood as interaction operators, experiments such as the thought experiment described above could in principle be possible. If it succeeds in realizing this successfully, one would of course have found the counter-example at hand at the same time. This also offers the opportunity to answer the question of whether an ultra-fast information transfer is possible.

The question as to whether information can be exchanged between quantum systems without resorting to a classical information channel thus does not depend essentially on the demands arising from the special theory of relativity to a relativistic quantum theory, but on the question of whether there are physically feasible non-local operators that can not be considered as interaction operators. On the question, which claims result by the special theory of relativity for a relativistic quantum theory, I would therefore not enter here. A good insight into current research on this question can be found in [52].

Even if one assumes that the classical idea is correct and thus a transmitter and a receiver are required for an information transmission and therefore must necessarily be transmitted from the sender energy to the receiver, it can not be concluded from this that an ultra-light information transfer is in principle impossible. For: The special theory of relativity is based on two postulates: I. All natural laws are covariant with respect to the transformations between inertial systems (all natural laws have the same form in all inertial systems). II.) The speed of light c has in all Inertialsystemen the same value. From the relativity principle (postulate I) it can only be deduced that there must be an invariant velocity with regard to the permissible transformations between inertial systems. That this invariant velocity, in whatever sense, represents an upper limit, can not be derived from the principle of relativity. Nor is it possible to deduce from the principle of relativity the concrete value for this invariant velocity. Only postulate II assigns a concrete value - the speed of light c - to this invariant velocity (see also [47], section 1). In order to be able to transmit a signal in the sense of classical physics, there must be a transmitter and a receiver of the signal. The transmitter must always transmit energy to the receiver for the purpose of signal transmission. This could be realized, for example, by the sender writing a message on a piece of paper wrapping it around a stone and then throwing it to the receiver. Of course, the transmitter could also send a suitably coded light signal to the receiver. However, since in the context of the special theory of relativity an acceleration of matter to a velocity ≥ c is not possible and the speed of light c has the same value in all inertial systems, a signal transmission with signals of this kind is obviously only possible at maximum with the speed of light. Since according to the principle of relativity but in principle also particles with a velocity> c (tachyons) are conceivable, it can not be ruled out in principle that by means of these hypothetical particles a signal transmission could also be possible with superluminal velocity (see also [46], section 3) , However, up to the present day there is neither a proposal for an experiment from which the existence of Tachyon could be derived, nor a proposal for an experiment that makes the fundamental feasibility of signal transmission by means of tachyons appear possible. Wheeler and Feynman explicitly pointed out that an over-light-fast causal effect does not necessarily lead to logical contradictions or paradoxes in the considered theory. [48] An example (a Lorentz-variant theory) was used. The argument that the assumption of an ultra-fast information transfer necessarily leads to temporal paradoxes or inconsistencies within the considered theory can therefore not be correct.

• VII: Einbindung des Spin-Statistik-Theorems in den Interpretationsansatz. Ich möchte hier der Frage nachgehen, welche Konsequenzen sich für den oben entwickelten Interpretationsansatz aus dem in der Quantenphysik gültigen Spin-Statistik-Theorem ergeben. Um diese Frage angehen zu können, ist es hilfreich, die in diesem Zusammenhang wesentlichen Begriffe und die dahinterstehenden Vorstellungen zuerst noch einmal näher zu betrachten.

As these considerations show, there is thus no fundamental argument from which it could be deduced that the thought experiment described above could not be realized.

• VII: Integration of the Spin Statistics Theorem in the Interpretation Approach. I would like to investigate the question of what consequences arise for the interpretation approach developed above from the spin-statistics theorem valid in quantum physics. To tackle this question, it is helpful to first take a closer look at the terms and the concepts behind them.

If one speaks of independent systems within the framework of classical physics, it is clear what they mean by this. Considering, for example, the classical electrodynamics, the systems can interact only by means of the electromagnetic fields. The resulting influence of one system on another system takes place in the physical space. As long as the two systems are close to each other, they can interact with each other almost instantaneously, because then the effect mediated by the electromagnetic fields occurs almost instantaneously. If the two systems are separated from one another and then enter, for example, at time T ₀ into two shielded, mutually separate spatial regions, then the systems can no longer interact from this point on, since an interaction is then excluded between the two systems. The systems can then be considered as independent from time T ₀ onwards. Depending on the way in which measurements are performed on the systems, however, correlations may occur between the measured values on one system and measured values on the other system. However, these can only emerge from the effect of the electromagnetic fields on the two systems, which is possible before time T ₀ . If these correlations are of a statistical nature, then the respective experiment merely has to be repeated sufficiently often under identical conditions in order to be able to reliably recognize these correlations. The cause of these correlations thus always lies in the past, since the measurements on the systems are always carried out at times T _M > T ₀ . Mathematically, this fact can be described by the Bell's inequalities (see for example [1]). The Bell's inequalities are met if and only if the statement "the systems are independent" is equivalent to the statement "there is no interaction between the systems". Thus, when speaking of "independent" systems in the context of classical physics, this is equivalent to the statement "there is no interaction between the systems".

Of course, in order to be able to speak of independent systems at the time of the measurement, one could just as well choose the distance between the considered areas of space so that these - based on the duration of the measurement can be regarded as a space-like separated. Of course, a shielding of the room areas would not be necessary.

If two independent systems A and B are considered within the framework of quantum physics, with the system half-dreams H _A and H _B , then a description of the overall system takes place in the Hilbert space H _BA = H _B × H _A formed over the tensor product of the system spaces. An operator E _A acting only on H _A is then assigned to H _BA the operator 1 × E _A (1 here is the identity operator). A localized on H _B E _B operator is then assigned to the operator H _BA E × _B. 1 As long as the systems can be regarded as independent, each state transformation can be described by an operator of the form E _B × E _A , ie as a tensor product of two operators acting locally on the respective system goal dreams. Interactions between the systems are therefore described on H _BA by operators W _BA , which can not be decomposed into a tensor product of two locally acting operators. Interaction operators W _BA are thus always non-local operators.

In quantum physics, however, the statement "there is no interaction between systems" is generally no longer equivalent to the statement "the systems are independent". In quantum physics, this equivalence applies only to quantum systems that can be described by a product state. But not for entangled quantum systems. This can be easily demonstrated by the following example:
In the EPR Bohm thought experiment (see ), the source Q emits two spin ½ systems in the singlet state Ψ ₃₁ (see Eq. Assuming that the source 0 does not emit the two spin ½ systems in the state Ψ ₃₁ , but in the state | + z, + z>, one assumes that the distance between the star Gerlach systems used as spin analyzers Magnets M _L and M _R is chosen so large that they, based on the duration of the measurement, can be regarded as spatially separated arranged, the following results: If the spin analyzers also z-axis quantization axis, so you get strictly correlated readings. Both spin analyzers will certainly deflect the spin ½ systems in the + z direction. If, for example, the quantization axis of the spin analyzer M _R arranged to the right of the source is rotated into the y direction, the measured values at the spin analyzer M _R can no longer be predicted. The systems are then deflected in the + y direction in 50% of the cases and in the -y direction in 50% of the cases. At the spin analyzer M _L , the systems will continue to be deflected with certainty in the + z direction. If one now also rotates the quantization axis of the spin analyzer M _L in the y-direction, individual measured values can no longer be predicted there either. Then in 50% of the cases the systems are also deflected in the + y direction and in 50% of the cases in the -y direction. However, it is then no longer possible to detect any correlations between the measured values of the two spin analyzers M _L and M _R.

Depending on the direction in which the quantization axes of the spin analyzers are pointing, the measured values between them are thus correlated to different degrees. However, as one can easily verify, the Bellschen inequalities are always fulfilled. The occurring correlations can therefore be interpreted for any orientation of the spin analyzers as a consequence of interactions in the preparation of the two systems in the state | + z, + z>. Since it was assumed that the spin analyzers were arranged spatially separated, an interaction of the systems at the time of measurement can be excluded with certainty. Thus, as in the case of classical physics, one can speak of "independent systems" since the statement "between systems there is no interaction" is equivalent to the statement "the systems are independent".

However, if the source Q emits two spin ½ systems in the singlet state Ψ ₃₁ , as assumed in the EPR-Bohm thought experiment, correlations occur between the measured values of the spin analyzers for certain orientations, which are not possible for interaction-free, independent systems. The Bellschen inequalities are then violated. The fact that the measured values of the spin analyzers for a quantum system in the singlet state Ψ _{31 can be} more correlated than this is possible for interaction-free, independent systems can also be seen in a very clear way for the singlet state. An essential property of state Ψ ₃₁ is (see, for example, [31]) that no quantization axis can be assigned to it. As long as the quantization axes of the spin analyzers used are always arranged in parallel, it does not matter in which direction they point. With respect to the quantization axes of the spin analyzers, the quantum system is then always in the singlet state. This property means that the readings of the spin analyzers, regardless of the direction of the quantization axes, are always strictly anti-correlated as long as the quantization axes are aligned in parallel. For interaction-free, independent systems, a correlation of this kind is not possible.

Entangled systems thus seem to possess properties that can no longer be assigned to the individual systems, but only to the overall system. Since these properties do not depend on how far the systems are from each other, nor are they affected by a shielding of the spatial areas in which measurements are taken on the systems, it is obvious to call these properties "internal properties" of the quantum system. Of course, this raises the question of how these internal properties of entangled quantum systems come about? The obvious assumption is then, of course, that there must be effects between the quantum systems. However, these can not be interactions. Because then the properties of the physical space, such as shielding or variations of the distance between the systems, would have an influence on the internal properties. Which is not the case. But this suggests the assumption that the internal properties of a quantum system can not exist in physical space. In this sense one could also speak of the fact that the inner qualities describe an indirect effect.

Of course, the fact that the intrinsic properties of entangled quantum systems are fundamentally new aspects of quantum physics and can not be understood as the cause of interactions between quantum systems can already be seen from the fact that they occur due to the supposition principle postulated in quantum physics.

If, however, the assumption is true that entangled quantum systems can be assigned inner properties, then of course the question arises whether internal properties can not generally be assigned to each quantum system? Then also quantum systems that are described by a product state would have internal properties. If the interpretation approach developed above applies, then this would be the case. Then all quantum systems would possess internal properties. There would be no isolated quantum systems. The difference between entangled quantum systems and independent quantum systems would then only be that certain intrinsic properties are accessible to entangled quantum systems, whereas for independent systems they are not accessible by measurement. Mathematically, this fact would then have to be reflected in the fact that properties of a quantum system are only directly accessible via the computing basis, but not via other basic systems such as the Bell basis. The calculation base would then have to be an excellent basic system in this sense. This would then be determined by the energetic representations possible for the quantum system under consideration (see Section III).

If the assumption is true that quantum systems generally have internal properties, then the question arises as to what role interactions play in the preparation of entangled quantum systems?

If one considers the preparation process described in Section III, in order to convert two ⁴⁰ Ca ⁺ ions (which are in a linear ion trap) starting from the product state | 1,1> into the singlet state, it is noticeable that for the formal Description of this state transformation, the degree of freedom of the common vibration of the two ions along the longitudinal axis of the linear ion trap is not at all necessary. Because: If one is not interested in a description of the specifically performed preparation _steps , the transfer of the product state | 1,1> into the singlet state can be formally described by the unitary operator U _B2B1 ^-1 (see equation 14).

But why is an interaction between the ions (mediated by the Coulomb repulsion) required? As shown in Section III, the degree of freedom of the common vibration of the two ions along the longitudinal axis of the ion trap is needed to ensure that it is in principle impossible to predict which one, which is possible for the singlet state of possible energetic representations, has actually been realized , The degree of freedom of the common vibration could thus also be understood as an aid, which indeed allows the process of entanglement, but which - analogous to a catalyst - emerges unchanged from the preparation process. Without wishing to overstate this comparison, the considerations outlined above suggest the question: is it also possible without a catalyst?

For this I would like the in and explain how it is described within the framework of the formalism of quantum physics and what conclusions are derived from the interpretive approach developed above. Two identically constructed sources Q1 and Q2 (are in not shown) should emit a photon always exactly at the same time. The source Q1 the photon 1 and the source Q2 the photon 2. The photon 1 is to impinge on the path a on a symmetrical, lossless beam splitter ST and the photon 2 is to impinge on the path b on the beam splitter ST. The paths a and b should meet on the beam splitter ST at the point P. Furthermore, the paths which the photons 1 and 2 have to travel from the sources Q1 and Q2 to the point of impingement P on the beam splitter should be exactly the same length. The source Q1 (Q2) should be arranged so that the photon 1 (Photon 2), if this is reflected at the beam splitter, on the way d (c) from the beam splitter again and the photon 1 (Photon 2) if this the beam splitter is transmitted, on the way c (d) away from the beam splitter again.

Furthermore, I would like to introduce the following notation: If a photon flies in the direction 1 (path a and c), the state | 1> _{i is} assigned to the photon i (i = 1,2). If the photon i flies in the direction 0 (path b and d), the state | 0> _{i is} assigned to it. Before the two independent, identical photons reach the beam splitter ST, then the state can = ₅₁ = | 0> ₂ | 1> ₁ = | 0,1> (51) be assigned. Applying now the beam splitter property (1) to the state Ψ ₅₁ , we obtain the expression Ψ ₅₂ = 1/4 ^1/2 (i | 1> ₂ | ₁ > ₁ + i | 0> ₂ | 0> ₁ + | 0> ₂ | 1> ₁ - | 1> ₂ | 0> ₁ ). (52)

Since the photons 1 and 2 have to be regarded as indistinguishable after they have passed the beam splitter, the expression Ψ ₅₂ for photons (generally for bosons) still needs to be symmetrized according to the spin statistics theorem [45] valid in quantum physics , This results in the state Ψ ₅₃ = i / 2 ^1/2 (| 1> ₂ | 1> ₁ + | 0> ₂ | 0> ₁ ). (53)

beschrieben werden. Unterscheidbare Systeme verhalten sich somit wie unabhängige, wechselwirkungsfreie Systeme. Können die beiden Photonen nach dem Strahlteiler nicht mehr als unterscheidbare Systeme betrachtet werden, so ist zur Beschreibung der Überführung des Zustandes |0,1> in den Zustand Ψ₅₃, im Unterschied zu dem lokalen Operator U_ST, ein nicht-lokaler Operator erforderlich, da die Anwendung des Spin-Statistik-Theorems zu einem verschränkten Zustand führt. In welchen Zustand die einlaufenden Photonen am Strahlteiler überführt werden, hängt somit offensichtlich von der Situation am Strahlteiler ab. Ob die Systeme nach dem Strahlteiler als unterscheidbar oder ununterscheidbar angesehen werden müssen. Können die Systeme nach dem Strahlteiler nicht mehr als unterscheidbar angesehen werden, so führt erst die Anwendung des Spin-Statistik-Theorems zu einem verschränkten Zustand.If these were not photons (bosons) but two fermions, then the expression Ψ ₅₂ would have to be anti-symmetrized. In the case of fermions, then the state would Ψ ₅₄ = 1/2 ^1/2 (| 0> ₂ | 1> ₁ - | 1> ₂ | 0> ₁ ) (54) result. The correct expression is thus obtained only after the expression Ψ _{52 has been} symmetrized in the case of bosons or anti-symmetrized in the case of fermions. Only the application of the spin statistics theorem leads to a correct description of the situation under consideration at the beam splitter ST. Only in the event that the two incoming quantum systems can still be regarded as distinguishable quantum systems even after the beam splitter ST (for example, if the two quantum systems hit the beam splitter in time so that the wave packets associated with the systems no longer spatially overlap), can the Expression Ψ ₅₂ already be understood as the, resulting at the beam splitter, state. For distinguishable systems, it does not matter if they are bosons or fermions. The state transformation effected by the beam splitter can then in both cases by the unitary operator describing the beam splitter property (1)

to be discribed. Distinguishable systems behave like independent, interaction-free systems. If the two photons can no longer be regarded as distinguishable systems after the beam splitter, a non-local operator is required to describe the transformation of state | 0,1> into state Ψ ₅₃ , in contrast to the local operator U _ST . because the application of the spin statistics theorem leads to an entangled state. The state in which the incoming photons are transferred at the beam splitter thus obviously depends on the situation at the beam splitter. Whether the systems after the beam splitter must be considered as distinguishable or indistinguishable. If the systems can no longer be considered as distinguishable after the beam splitter, then the application of the spin statistics theorem leads to an entangled state.

However, this raises the question of what effect can be assigned to the application of the spin statistics theorem? To address this question, let me suppose that there is a unitary operator U _NLB , such as the effect of applying the spin statistics theorem by means of the unitary operator U _ST in the form Ψ ₅₃ = U _NLB U _ST | 0.1> (56) can describe. But what form does the operator U _{NLB have} ? To be able to determine this, I would like to rewrite the state Ψ ₅₂ . With regard to the Bell base you get: Ψ ₅₂ = 1/4 ^1/2 (i | 1> ₂ | 1> ₁ + i | 0> ₂ | 0> ₁ + | 0> ₂ | 1> ₁ - | 1> ₂ | 0> ₁ ) = 1 / 2 ^1/2 (iΦ ₊ - Ψ _- ) = e ^iπ / 2 ^1/2 (e ^{-iπ / 2} Φ ₊ + Ψ _- ) = e ^iπ / 2 ^1/2 (| A> + | B>). (57) With: | A> = e ^{-iπ / 2} Φ ₊ and | B> = Ψ _- , where: ^<A | ^B> = 0.

so erhält man: U_NLBΨ₅₂ = e^iπ|A> = iΦ₊ = Ψ₅₃ (59) The state Ψ ₅₂ can, however, also be formally understood as an element of a two-dimensional vector space spanned by the orthogonal state vectors | A> and | B>. If we define the unitary operator U _NLB according to this vector space

so you get: U _NLB Ψ ₅₂ = e ^iπ | A> = iΦ ₊ = Ψ ₅₃ (59)

For the considered situation at the beam splitter, the effect of the application of the spin statistics theorem on the state Ψ ₅₂ for bosons can thus be formally described by means of the operator U _NLB . In the case that the two quantum systems are not photons (bosons) but fermions, you get immediately: U _NLF Ψ ₅₂ = e ^iπ | B> = -Ψ _- = Ψ ₅₄ (60) With: U _NLF = (U _NLB ) ^-1 (61)

U _NLF is thus given by the operator inverse to U _NLB .

Which effect can be assigned to the operators U _NLB and U _NLF, and thus to the application of the spin statistics theorem, can now be directly deduced from the properties of the state Ψ ₅₂ in Eq. (57). The state Ψ ₅₂ is a product state of the form Ψ ₂₀ , with: φ = π / 2 and θ = π. These have the property that the energetic representations assigned to the quantum system in state Ψ ₅₂ correspond precisely to those that can be assigned to the quantum system via the internal properties relating to the Bell basis (see also Section III). However, the internal properties associated with the Bell base of the quantum system in state Ψ ₅₂ over the state components Φ ₊ and Ψ _- must thus be understood as properties physically realized for the quantum system. However, since the properties assigned to the state components Φ ₊ and Ψ _- can only be regarded as properties of the overall system, the state Ψ _{52 can} also be understood as an overlay state in which the subsystems of the quantum system are entangled via the state components Φ ₊ and Ψ _- . The subsystems can not therefore be regarded as energetically isolated systems (see also Section III). However, since this does not result in metrologically recognizable correlations, the subsystems can be regarded as independent systems. However, in the situation at the beam splitter for indistinguishable systems in state Ψ _{52, one} can not say that the quantum system already has an energetic representation. Indistinguishable quantum systems in state Ψ ₅₂ can only be understood as elementary quantum systems in the situation at the beam splitter. Distinct quantum systems in state Ψ ₅₂ , on the other hand, are always energetically represented quantum systems.

The effect of the operator U _NLB (U _NLF ) can thus be seen in the fact that this, the state of superposition Ψ ₅₂ consisting of the state components Φ ₊ and Ψ _{- converted} into a state by the state component Φ ₊ (Ψ _- ) is fixed and the state vector Φ ₊ (Ψ _- ) is retained as a physically realized property of the quantum system. That this is a quite remarkable property becomes clear when one considers, for example, the properties of the state transformations described by the operators U _NLB and U _NLF with those of the state transformation U _B2B1 ^-1 discussed above, which the conversion of the product state | 1,1> into the Singlet state describes, compares. In both cases, a product state is converted into an entangled state. However, while the operator U _NLB (U _NLF ) obtains the state component Φ ₊ (Ψ _- ) as a physically realized property, in the case of the operator U _B2B1 ^{-1 it} can not be said that it receives a state vector as an already physically realized property. Because: The product state | 1,1> can not be assigned the state part Ψ _- as a physically realized property, nor can the product state | 1,1> be interpreted as a superposition state with respect to the Bell basis, since the state in the Bell basis associated state portions Φ ₊ and Φ _{- can be assigned} only as inherent properties but not as physically realized properties.

The effect associated with the operator U _B2B1 ^-1 thus differs quite fundamentally from the effect that can be attributed to the operators U _NLB and U _NLF . This naturally suggests that the preparation _procedure assigned to the operator U _B2B1 ^-1 must also fundamentally differ from the preparation _procedure assigned to the operators U _NLB and U _NLF . But in what can the main difference be seen?

It is clear that in both cases the preparation processes have to be designed in such a way that it is in principle impossible to predict which energetic representation is actually realized. The main difference lies in how this goal is achieved. While in the case of the operator U _B2B1 ^-1 the preparation method used is based on the fact that individual ions can be manipulated specifically by means of the manipulation _laser (see Section III), this is obviously impossible in the situation at the beam splitter in principle. At the beam splitter you have neither the possibility to perform specific preparation steps on individual subsystems, nor the possibility to influence the process as such in any way. By this is meant that, in contrast to the state preparation by means of the manipulation laser, in which the resulting state directly from the properties of the laser light of the manipulation laser, such as the correct choice of intensity, the Puldauer and the relative phase of the laser pulse to the oscillating electrical moment of illuminated ⁴⁰ Ca ⁺ ions depends, there are no parameters in the situation at the beam splitter in principle, on the process of state preparation may be affected.

Exactly this fact seems to me to be the essential point, since it makes it possible to transfer the concept of "indistinguishability" essential for the application of the spin-statistics theorem to preparation processes and to generalize it in the form of the postulate (P2):
P2): If a state of the form Ψ _{52 can be} assigned to a quantum system consisting of two identical bosons (fermions) for the considered degree of freedom, then the operator U _NLB (U _NLF ) can always be physically realized if and only if the quantum system for the considered one Degree of freedom has no energetic representation and there is a preparation process that transforms the quantum system into an energetically represented quantum system for the relevant degree of freedom and fulfills the following conditions: (P2.1): In principle it must be impossible to manipulate individual bosons (fermions) can and (P2.2): In principle, it must be impossible to influence the process as such in any way. Each preparation process that satisfies conditions (P2.1) and (P2.2) then converts state Ψ ₅₂ to state Ψ ₅₃ (Ψ ₅₄ ).

Based on the preceding considerations, it is clear that postulate (P2) correctly describes the situation at the beam splitter. But what is the generalization? The essential point is that (P2) can be formulated without having to refer to the term "indistinguishable". In contrast to the spin statistics theorem, (P2) does not explicitly require that the quantum systems involved must be indistinguishable. However, it can not be concluded from this that Postulate (P2) does not implicitly include the requirement for the indistinguishability of the systems. However, at least for spin 1/2 systems, it is possible to check, using the thought experiment described below, whether conditions (P2.1) and (P2.2) in postulate (P2) implicitly include the requirement for the indistinguishability of the systems involved:
I would like to consider two identical spin 1/2 systems that are at rest. The distance between these should be so large that they can not interact. Furthermore, I would like to assume that the two systems can be superimposed with a homogeneous magnetic field B _z in the z-direction. This should be able to be turned on and off as desired. When the magnetic field B _z is activated, the energy difference between the two energetic eigenstates of the spin 1/2 systems is given by ΔE _z = 2μB _z (see Section IV). The energetically lower energy eigenstate of the ith system (i = 1, 2) is given by | 1> _i and the higher energy eigenstate is denoted by | 0> _i . At the beginning, the magnetic field is switched off and the quantum system consisting of the two Spin 1/2 systems should be present in the product state Ψ ₅₂ . Since the two energy levels coincide when the magnetic field is switched off, the quantum system has no energy representation. The quantum system must therefore be considered as an elementary quantum system. Now, however, the transfer of an elementary quantum system into an energetically represented quantum system can be considered as a preparation step. This can be realized by placing the quantum system in a magnetic field or by switching on a magnetic field (see also Section IV). For the activation and deactivation process of the homogeneous magnetic field B _z , the boundary condition (R3) should be satisfied: Δt _S <t _max = h / (4πΔE _z / 2), (R3) where Δt _{S is} the time required to turn the magnetic field on or off, with h the Plank's effect quantum and with t _max the energy-time uncertainty relation Eq. (29) given maximum possible time is designated (see also section V).

If one switches the magnetic field B _z is now so that the boundary condition (R3) is satisfied, so full of, by the switching-on of the homogeneous magnetic field B _z realized to preparation step realize the required in (P2) conditions to the operator U _NLF physico , Because: Condition (P2.1) is met, since it was assumed that the magnetic field B _{z is} homogeneous and thus always affects the same magnetic field on both spin 1/2 systems at any given time. Condition (P2.2) is fulfilled, since the boundary condition (R3) ensures that the energetic eigenstates of the systems during the switch-on process (ie for times <t _max ) no defined energy and therefore no temporal state development can be assigned to the quantum system. As a result, it is impossible to influence the preparation process in a targeted way. Since the conditions necessary for the applicability of postulate (P2) are fulfilled, (P2) can be applied to the thought experiment. According to postulate (P2), the quantum system must therefore be present as energetically represented quantum system in the state Ψ _- after the switch-on process.

The thought experiment can most easily be realized by means of the microchip ion traps described in [51]. These miniaturized ion traps allow the magnetic field superimposed on the ion trap to be generated via sufficiently low-inductive air coils in the form of a Helmholtz arrangement. In this way it can be ensured that the homogeneous magnetic field superimposed on the ion trap can be switched on and off sufficiently quickly in accordance with the boundary condition (R3). In order to carry out the thought experiment, a ⁴⁰ Ca ⁺ ion is captured in two identical ion traps in a first step. The qubits are realized via the energy levels | 1> and | 0> of the ⁴⁰ Ca ⁺ ions (see ). Then the quantum system consisting of the two ions is prepared according to the method described in Section III in the state Ψ ₅₂ . For this purpose, it is assumed that the two ion traps each have an (identical) magnetic field B _z superposed with the z axis as the quantization axis. However, in order to perform the thought experiment, the state of the quantum system must be represented by the energy levels | 1> and | 2> (see also ). This can be achieved via a suitable π-pulse (see also Section IV). The formal state of the whole system is not affected by this preparation step (see section IV). After the ions have been prepared in state Ψ ₅₂ , the magnetic fields superimposed on the individual ion traps are to be switched off one after the other. In this case, the magnetic fields superimposed on the ion traps should be switched off in such a way that state Ψ _{52 is} maintained. This can be realized, for example, by switching off the magnetic fields in compliance with the boundary condition (R3) and selecting the different switch-off times so that the process of coherently coupled vacuum fluctuation can certainly not occur (see the remarks below) and for the state parts of the quantum system give the correct phase (see also Section VI). The quantum system then exists as an elementary quantum system in state Ψ ₅₂ . If the magnetic fields B _z which can be superimposed on the ion traps are again switched on at the same time after a certain time according to the boundary condition (R3), the quantum system should be converted into the energetically represented state Ψ ₅₄ , provided that the postulate (P2) is correct. In order to apply the state analysis procedure described in Section III, the state of the quantum system must be represented by the energy levels | 0> and | 1>. This can again be achieved by a suitably chosen π-pulse (see Section IV).

An important application could be for the thought experiment in the realization of an interaction-free entanglement of quantum systems in quantum computers. Although the thought experiment using micro-ion traps can be easily performed, these are probably unsuitable for quantum computer, since the technical complexity is likely to be too large. The most promising for quantum computers Candidate could be diamond, as diamond allows the technical realization of quantum bits at room temperature. Corresponding experiments are described, for example, in [50]. In diamond single quantum bits can be realized by ¹³ C isotopes contained in natural diamond with about 1%. ¹³ C has one, in contrast to ¹² C, non-zero nuclear spin (S = 1/2). These can be read or addressed by NV centers (see [50]). If the thought experiment described above can also be carried out with quantum bits which are realized by ¹³ C atoms in diamond, this would be an important step towards a quantum computer which can be realized technically and with reasonable effort.

If the thought experiment can be carried out successfully, at least for spin 1/2 systems it would be shown that postulate (P2) can also be applied to distinguishable quantum systems. However, the thought experiment is also of fundamental importance with regard to another aspect. Due to the selected boundary conditions, there can be no interaction between the two Spin 1/2 systems. The non-local operator U _NLF describing the state transformation can therefore not be understood as an interaction operator. So if the thought experiment can be successfully carried out, there must also be effects between independent quantum systems. If true, independent spin 1/2 systems can not be considered as isolated quantum systems, and the interpretation (IN5) must be valid for at least Spin 1/2 systems.

Looking more closely at the thought experiment described above, it is noticeable that under the boundary condition (R3) assumed for the thought experiment, the process of coherently coupled vacuum fluctuation postulated in postulate (P1) can always occur. The operator U _NLF could therefore also be interpreted as describing the process of coherently coupled vacuum fluctuation. If this interpretation were correct, this would mean that coherently coupled vacuum fluctuations under these conditions occur with certainty after postulate (P2). The reason for this could then also be seen in the fact that the state vector Ψ _{- is preserved} as a physically realized property of the quantum system.

Since according to postulate (P1) for the state describing the quantum system only that it must contain the state component Ψ _- , the thought experiment described above should then also be able to be carried out if the initial state is not the state Ψ ₅₂ but generally a state of Form Ψ ₂₀ with arbitrary (real) phases φ and θ, is chosen as an elementary state for the quantum system.

Write the product condition Ψ ₂₀ again according to Ψ ₂₀ = e ^iθ / 2 ^1/2 (1/2 ^1/2 (| 0,0> - e ^i2φ | 1,1>) + e ^iφ Ψ _- ) = e ^{i (θ + φ)} / 2 ^{1 / 2} (e ^-iφ / 2 ^1/2 (| 0,0> - e ^i2φ | 1,1>) + Ψ _- ) = e ^{i (θ + φ)} / 2 ^1/2 (| A> + | B> ) (62) With: | A> = e ^-iφ / 2 ^1/2 (| 0,0> - e ^i2φ | 1,1>) and | B> = Ψ _- , where: ^<A | ^B> = 0, um, on the Hilbert space spanned by the vectors | A> and | B>, analogous to Eq. (61) and Eq. (58) an operator U _NLF can be defined for which: U _NLF (Ψ ₂₀ ) = e ^{i (θ + φ)} | B> = e ^{i (θ + φ)} Ψ _- (63)

Of course, this formally has the same form as the operator U _NLF in (P2) (see equation (61) in connection with equation (58)).

Thus, if the state Ψ ₂₀ with arbitrary (real) phases φ and θ, as the elementary state for the quantum system, is selected as the initial state in the thought experiment described above, and then the magnetic field B _{z is switched} on taking into account the boundary condition (R3), then Quantum system, except for the global, unobservable phase e ^{i (θ + φ)} , then in the energetically represented state Ψ _- present.

The thought experiment also offers the opportunity to examine the question already discussed in Sections V and VI as to whether the distance between the subsystems, between which a coherently coupled vacuum fluctuation can occur (here the two Spin 1/2 systems), can be arbitrarily large, or whether this is limited to the top.

• VIII. Philosophische Konsequenzen: Die sich aus dem im Abschnitt VI vorgeschlagenen Gedankenexperiment ergebenden Konsequenzen wären allerdings nicht nur auf unser physikalisches Weltbild beschränkt. Bei vielen Wissenschaftlern scheint sich die Vorstellung durchgesetzt zu haben, dass es so etwas wie Bewusstsein als eigenständige Eigenschaft nicht geben kann (siehe beispielsweise [39], [40], [41]). Diese Vorstellung beruht in der Regel auf der philosophischen Position des ontologischen Materialismus. Der Materialismus geht davon aus, dass es auch für eine Eigenschaft, wie etwa das Bewusstsein, eine materielle Basis geben muss. Bei geeignet gewählter materieller Basis, kann dann aus physikalischen Wechselwirkungen Bewusstsein hervorgehen. Bewusstsein ist damit lediglich die Folge dieser physikalischen Wechselwirkungen. Damit es so etwas wie Bewusstsein geben kann, muss es somit eine materielle Basis geben, auf der dann das Bewusstsein entstehen kann. Bewusstsein oder auch ganz allgemein mentalen Prozessen wird somit eine passive Rohe zugeordnet, die bestimmte Gehirnaktivitäten lediglich „begleiten”. Man spricht dem Bewusstsein damit aber jede kausale Wirkung ab. Doch auf welchen Erkenntnissen beruhen diese Vorstellungen? Als Begründung wird lediglich darauf verwiesen, dass es keinerlei Hinweise gibt, dass es so etwas wie Bewusstsein als eigenständige Eigenschaft geben könnte. Berücksichtigt man dann noch die Tatsache, dass es bis heute noch nicht einmal möglich ist, eine brauchbare Definition oder wenigstens konkrete Vorstellungen, hinsichtlich der für mentale Prozesse erforderlichen Voraussetzungen gibt, so kann man diesen Standpunkt nicht wirklich als überzeugend ansehen.

I would like to briefly discuss another application for the micro-ion traps described in [51]. The thought experiment described in Section VI can also be realized by means of two micro-ion traps. For this purpose, two ⁴⁰ Ca ⁺ ions are trapped in each ion trap. The qubits are realized via the energy levels | 1> and | 0> of the ⁴⁰ Ca ⁺ ions (see ). In the ion trap 1, the ions S ₁ and S ₂ and in the ion trap 2, the ions S ₃ and S _{4 are} located. Both ion traps are superimposed on an (identical) magnetic field B ₂ . In each ion trap, the two localized ions are then prepared into the singlet state, so that the state Ψ _- ^43/21 results again for the state of the entire system. In order to be able to prepare the ions located in the individual ion traps in the singlet state, the distance of the ions in the ion traps should initially be chosen so small that they can interact by means of Coulomb repulsion and realized the preparation method described in Section III can be. However, in order to perform the thought experiment, the state of the quantum system must be represented by the energy levels | 1> and | 2> (see also ). This can be achieved via a suitable π-pulse (see also Section IV). Now, the distance of the ions in the traps only has to be increased to a few mm to neglect the Coulomb repulsion between the ions. For the magnetic fields superimposed on the two ion traps, I would like to assume that these are limited to the spatial region of individual ions and can be switched on and off locally while observing the boundary condition (R3). It should therefore be possible, in the one assigned to the individual ion space regions, the magnetic fields B _z switched on and to switch off, without thereby affecting the the respective other space portions (each also contain an ion) superimposed magnetic fields B are influenced _z. Variant 2 of the thought experiment described in Section VI could now be realized simply by simultaneously switching off the magnetic fields B _z in the regions containing the ions S ₁ and S ₄ while observing the boundary condition (R3) and at suitably selected times , again in compliance with (R3), the magnetic fields are switched on again after a time delay (see section VI). Then the quantum system should be present as an energetically represented quantum system in state Ψ ₄₉ , if the thought experiment in variant 2 can be successfully realized. In order to apply the state analysis procedure described in Section III, the state of the quantum system must be represented by the energy levels | 0> and | 1>. This can again be achieved by a suitably chosen π-pulse (see Section IV). Analogously, of course, the variant 1 of the thought experiment can be realized.

• VIII. Philosophical Consequences: The consequences of the thought experiment proposed in Section VI, however, would not be limited only to our physical world view. For many scientists, the notion that there is no such thing as consciousness as an independent property seems to have prevailed (see, for example, [39], [40], [41]). This idea is usually based on the philosophical position of ontological materialism. Materialism assumes that there must also be a material basis for a quality, such as consciousness. With a suitably chosen material basis, then physical interactions can give rise to consciousness. Consciousness is thus merely the consequence of these physical interactions. In order for there to be something like consciousness, there must be a material basis on which consciousness can emerge. Consciousness or even more general mental processes is thus assigned a passive raw, which only "accompany" certain brain activities. However, the consciousness is thus denied any causal effect. But on what findings are these ideas based? The reason given is merely that there is no evidence that there could be something like consciousness as an independent property. If one takes into account the fact that it is not even possible to date, or a viable definition, or at least concrete ideas, regarding the conditions necessary for mental processes, then one can not really consider this point of view convincing.

An entirely different idea underlies the philosophical position of interactionist dualism. This assumes that in nature there are properties or manifestations (entities) that are in principle inaccessible through physical interactions (intangible entities) and those that are accessible through physical interactions (material entities), where between intangible entities and material entities causal interaction is possible. The position of interactional dualism, of course, raises the question of how consciousness can affect the brain and the brain when there is no physical interaction between consciousness and the brain. The neurobiologist Sir John Eccles and the quantum physicist Friedrich Beck have proposed an interesting hypothesis ([39], chapter 9, page 216):
"... that the mental intent of the self becomes neuronally effective by temporarily increasing the probabilities for exocytosis in a whole dentron and thus coupling the large number of probability amplitudes to achieve a coherent effect ( 6.10 and 9.2 ). "

Of course, a quantum-physical description of the exocytosis is required. Exocytosis is the opening of a channel in the presynaptic vesicle lattice (PVG) and the discharge of the vesicle transmitter molecules into the synaptic cleft. The opening of a channel in the PVG is caused by a nerve impulse that continues into a bouton (axon terminal). Although a large amount of Ca ²⁺ ions flow into the axon during the transmission of a nerve impulse in an axon, it is assumed that by attaching four Ca ¹⁺ ions to a synaptic vesicle, a channel immediately opens through the adjacent presynaptic membrane so that all of its transmitter content is released into the synaptic cleft. Eccles and Beck now postulate that this triggering mechanism results from quantum transitions between metastable molecular states mediated by quasi-particles ([39], Chapter 9, page 226):
"... preparation for exocytosis means that the paracrystalline PVG is placed in a metastable state in which exocytosis can occur. The triggering mechanism is then presented as a movement of a quasiparticle with one degree of freedom along a collective coordinate and across an activation barrier ( ). This movement is due to a quantum mechanical tunneling process through the barrier (similar to radioactive decay) ".

The probability that the quasi-particle can tunnel through the activation barrier can, it is believed, now be felt by the consciousness (the self). Consciousness would thus act on the transmission of signals at the exact locations in the brain where the electrical signals are briefly converted into chemical signals by means of the vesicle transmitter molecules.

Strictly speaking, however, Eccles and Beck say nothing about how the effect of consciousness on the considered quantum systems comes about. It also remains unclear whether the effect of consciousness on the quantum systems under consideration can be described within the formalism of quantum physics and, if so, how the formalism of quantum physics must be interpreted. But this point is decisive for the acceptance of this hypothesis. Since the considered processes are anything but easy-to-calculate situations, it would be extremely difficult to verify this hypothesis. The fact that this hypothesis is at least principally verifiable characterizes the position taken by the interactionalist dualism with regard to the position of materialism. Because for the position of materialism, there are not even rudimentary verifiable hypotheses.

If one assumes that it is not possible with the current state of the art to examine the hypothesis proposed by Eccles and Reck, then what possibilities could there be to decide which philosophical position is correct? The position of materialism or that of interactionalist dualism? Of course there are many other philosophical positions [39]. But I would not like to go into that here, because the fundamental questions can be clearly shown by means of these two positions.

The essential reason for the rejection of the position of interactionalist dualism, especially of scientists who represent the position of materialism, seems to me to be the following point:
If the position of interactionalist dualism is correct, the physical (material) world can not be causally closed. There must then be physical events that are causally related, but the causal relationship can not be described through physical interactions.

A strong indication that the position of the interactionalist dualism is correct would therefore be an experiment that can only be interpreted as meaning that the physical (material) world can not be causally closed.

The thought experiment proposed in Section VI offers this possibility. If, in the thought experiment described there, the readings available to Alice and Bob depend on whether sources Q _2/1 and Q _3/4 are operated synchronously or asynchronously, this would be a clear indication that the physical world is not causally closed can. The main argument against the position of interactionalist dualism would then be refuted.

Finally, I would like to point out possible connections that could exist between the considerations outlined above and certain human abilities that are repeatedly described in Buddhist literature. It should be noted, however, that there are many different directions within Buddhism due to the different traditions of the particular regions in which Buddhist practice spread. However, I would like to comment on these differences here do not go down. One of the frequently described human abilities is that of conscious (lucid) dreaming (see, for example, [42]). The time the body sleeps is considered a way to train consciousness. From this results also the special meaning, which one assigns to the practice of the dreaming. If the consciousness is sufficiently trained, it is possible that two people in the dream can exchange information, even if they are in far away places. This ability is assigned to consciousness, which is conceived as an immaterial entity. Physical interactions are therefore not required for this information transfer in the dream.

From the standpoint of materialism, this assertion is simply impossible, since consciousness is conceived as a consequence of physical interactions. Thus, the two persons would have to interact via physical interactions. But what can be excluded with certainty, if the distances are only large enough, or it is ensured that no physical interactions can occur. From the standpoint of interactionist dualism, however, information transmission in the dream could certainly be possible. For this, one would have to assume, however, that there are structures in the brain that enable an information transfer analogous to the experiment proposed in Section VI. Whether these structures agree with the paracrystalline PVG (or single channels in the PVG) underlying Eccles and Beck's hypothesis, and whether, in contrast to coherently coupled vacuum fluctuations, as assumed by Eccles and Beck, must be used instead of the postulated quasiparticles Remain a question.

Assuming that the statements made in the Buddhist literature (see, for example, [42], [43], [44]) about the properties of consciousness are correct, it is natural to assume that all properties attributed to consciousness, such as the Ability to perceive, or the ability of the will, in the sense of interpretation (IN5) and (IN6), to exert their effect on the quantum system formed by the brain, by the quantum systems formally equivalent to this quantum system and the quantum system, realizable energetic representations. It is obvious, then, that all the information in the brain that we have acquired through our sensory organs throughout our lives forms the quantum system formed by the brain in such a way that the degree of energetic isolation of energy that is possible for that quantum system is other quantum systems in the sense of interpretation (IN5) and (IN6), and thus can be shaped to a certain extent by appropriate practice. The description of the physical world, based on the data of our sense organs, but also the perceptible thoughts, would then be based on physical processes in the brain, which then can perceive the consciousness. Thus, the physical processes in the brain, the consciousness (as described by the quantum physical state of this quantum system) would act and the consciousness in turn affect the physical processes in the brain, without this physical interactions are involved.

If one succeeds in verifying these assumptions, it would prove that the effect of the properties attributed to consciousness on the physical world, but also the effect of the physical world on the consciousness, by means of the interpretation approach proposed here, within of the formalism of quantum physics. I would like to discuss the related issues in more detail elsewhere.

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QUOTES INCLUDE IN THE DESCRIPTION

This list of the documents listed by the applicant has been generated automatically and is included solely for the better information of the reader. The list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions.

Cited non-patent literature

• Facts 1935 Albert Einstein, Boris Podolsky and Nathan Rosen (EPR) [0035]

Claims (7)

Method for the interaction-free entanglement of quantum systems in quantum computers, characterized in that the quantum systems to be entangled are prepared in a first preparation step in the state Ψ ₂₀ with arbitrary (real) phases φ and θ as elementary quantum system and then in a second preparation step by switching a homogeneous Magnetic field B _z , which includes the space regions in which the quantum systems to be entangled are located and in compliance with the boundary condition (R3) in the entangled state Ψ _- transferred. A method according to claim 1, characterized in that the quantum systems used are realized by ¹³ C atoms in diamond. A method according to claim 1, characterized in that the quantum systems used are realized by ⁴⁰ Ca ⁺ ions in the electronic ground state (4 ² S _1/2 ). Method according to Claim 2, characterized in that the quantum systems are addressed and / or read out by means of NV centers. Method according to one of claims 1 to 4, characterized in that the homogeneous magnetic field B _{z is} realized by means of low-inductance coils in the form of a Helinholtz arrangement. Method according to one of claims 1 to 5, characterized in that the entangled quantum systems are used to transfer the state of a quantum system to another quantum system and this preferably the method of teleportation is used. Method according to one of claims 4 to 6, characterized in that the quantum systems used are arranged on a rotating, preferably round, sheath so that these, in analogy to the magnetic districts of a magnetic hard disk, can be brought into a defined time in areas, which a homogeneous magnetic field B _z can be superimposed.

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