DE102019008600A1 - Process for the interaction-free transfer of information using entangled pairs of photons - Google Patents

Abstract

The invention relates to a method for interaction-free information transmission by means of entangled photons, in which a source (Q1) simultaneously generates two photons (P1) and (P3) entangled in at least one secondary degree of freedom and a source (Q2) simultaneously entangles two in at least one secondary degree of freedom Photons (P2) and (P4) generated. The transmitter of the information (Alice) can transmit the binary-coded information by means of a measurement of the first type (MT0) to the photons (P1) and (P2) the quantum system consisting of the photons (P3) and (P4) in a mixed state of the first type (Z1) and thereby encode the first logic state, preferably the logic 0, and by means of a measurement of the second type (MT1) on the photons (P1) and (P2) that of the photons (P3) and (P4 ) convert existing quantum system into a mixed state of the second type (Z2) and thereby encode the second logical state, preferably the logical 1. A symmetrical phase grating (SPG), preferably a symmetrical reflection grating, is used to implement the measurement of the first type (MT0). The receiver of the information (Bob) analyzes the state of the photons (P3) and (P4) with regard to the property mixed state of the first type (Z1) and mixed state of the second type (Z2) and can use this to transfer the logic states for the purpose recognize an interaction-free transfer of information. The secondary degree of freedom can advantageously be realized by means of the degree of polarization freedom and / or the degree of energy freedom. A beam splitter, preferably a symmetrical beam splitter, can be used for the state analysis of the photons (P3) and (P4).

Description

The invention relates to a method for the interaction-free transmission of information by means of entangled pairs of photons.

The method described in [1] is known for the interaction-free transfer of information (see also [2], where this is described again in detail and discussed in detail). A transfer of information without an exchange of energy (without an interaction) is not possible within the framework of classical physics. This is only conceivable in the context of quantum physics using entangled states. In [1], the entanglement is carried out in such a way that it enables interaction-free information transfer. For this purpose, several quantum bits (qubits) of the transmitter and the receiver are prepared beforehand in a suitable initial state. These can then be entangled by superimposing a defined magnetic field on the qubits to be entangled sufficiently quickly [1], [2]. The main disadvantage of this method, however, can be seen in the fact that in this method the processes of the transmitter and the receiver required for the entanglement have to be synchronized very precisely in time. In practice, this is very complex and involves high costs.

A method for the interaction-free transfer of information using entangled photons is also known [24]. This offers the possibility of transmitting information without interaction by means of two entangled pairs of photons. For this purpose, the transmitter can carry out two different measurements on the accessible photons. The receiver can tell which measurement was carried out by the photons accessible there. The main disadvantage of this method, however, can be seen in the fact that the required structure can only be implemented technically with great effort and is therefore very expensive.

Proceeding from this, the object of the invention is to improve the method for interaction-free information transmission described in [24] in such a way that the proposed method can be implemented with simple means and only commercially available components can be used.

The invention is based on the knowledge that, with sufficiently complex, entangled quantum systems, there are several possibilities for a measurement on a subsystem, which change the principally accessible knowledge of the entangled quantum system in a clearly distinguishable manner. This fact opens up the possibility of a freely selectable, interaction-free state preparation of the quantum system. A state preparation and state analysis using symmetrical beam splitters then enables an interaction-free transmission of information.

To achieve this object, the combinations of features specified in claims 1 to 7 are proposed. Advantageous refinements and developments of the invention result from the dependent claims 2 to 7.

In order to better understand the procedure proposed here for the interaction-free transfer of information by means of entangled photons, it is helpful to briefly consider some very basic relationships. For this purpose, the terms and designations used in [2] are adopted and, as in [2], apart from places where explicitly speaking of mixed states, only pure states are considered.

Entangled quantum systems can occur due to the superposition principle postulated in quantum physics and supported by a large number of experiments. The superposition principle is understood to mean the following (see, for example, [3] and [4]): If there are several different options when preparing a condition, it is in principle impossible to decide which option has been realized the state resulting from the preparation process is determined by the sum (with a countable number) of the individual possibilities, weighted with the respective probability amplitude.

Furthermore, in the context of quantum physics for a closed quantum system, it is required that the temporal development (the state dynamics) must be describable using unitary operators. This of course also applies to the state dynamics, which can be assigned to a preparation process as long as it receives the standard. Preparation processes that receive the standard must therefore be carried out using unitary operators can be described. Operators that can be understood as a description of a concrete, physical process are referred to below as "physically realizable" operators.

Independent quantum systems can always be described by product states in the context of quantum physics. Consider two independent quantum systems A and B , with the system help dreams H _A and H _B , a description of the overall system takes place in the Hilbert space H _BA = H _B × H _A formed by the tensor product of the system rooms. An operator E _A the only on H _A acts on H _BA then the operator 1 × E _A (1 is the identity operator here). One on H _B localized operator E _B is then on H _BA assigned the operator 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 , i.e. as a tensor product of two operators acting locally on the respective system auxiliary spaces. Interactions between the systems are therefore based on H _BA through operators W _BA described, which can not be broken down into a tensor product by two locally operating operators. Interaction operators W _BA are therefore always non-local operators.

• Die betrachtete Anordnung ist schematisch in dargestellt.

However, it cannot be concluded from this that every physically realizable, non-local, unitary operator describing a state transformation describes an interaction and must therefore be understood as an interaction operator. An example shows that there are also physically realizable, non-local, unitary operators describing a state transformation that cannot be interpreted as interaction operators:

• The arrangement under consideration is schematic in shown.

Two identically structured sources Q1 and Q2 should emit one photon each at freely selectable times. The source Q1 the photon 1 and the source Q2 the photon 2nd . The photon 1 should be on path a to a symmetrical, lossless beam splitter ST hit and the photon 2nd should be on path b to the beam splitter ST hit. The ways a and b are supposed to be on the beam splitter ST at the point P to meet. Next are the ways that the photons 1 and 2nd from the sources Q1 and Q2 to the point of impact P have to travel on the beam splitter, be exactly the same length. The source Q1 ( Q2 ) should be arranged so that the photon 1 (Photon 2nd ), if this is reflected on the beam splitter, on the way d (c) removed from the beam splitter and the photon 1 (Photon 2nd ) when this transmits the beam splitter, on the way c (d) removed from the beam splitter (see ).

zugeordnet werden. Die, durch den symmetrischen Strahlteiler bewirkte, Zustandstransformation lässt sich dann für zwei unterscheidbare Photonen (dieses kann beispielsweise dadurch erreicht werden, wenn die beiden identischen Photonen zeitlich so versetzt auf den Strahlteiler ST auftreffen, dass die den Photonen zugeordneten Wellenzüge (Wellenpakete) am Strahlteiler nicht überlappen) durch den unitären, lokalen Operator U_ST beschreiben [5], [6]: $${\text{U}}_{\text{ST}}=\frac{1}{\sqrt{4}}\begin{array}{c}|\mathrm{0,0}\rangle \\ |\mathrm{0,1}\rangle \\ |\mathrm{1,0}\rangle \\ |\mathrm{1,1}\rangle \end{array}\begin{array}{c}\begin{array}{cccc}\langle \mathrm{0,0}|& \langle \mathrm{0,1}|& \langle \mathrm{1,0}|& \langle \mathrm{1,1}|\end{array}\\ \left[\begin{array}{cccccccc}& 1& & \text{i}& & \text{i}& & \text{-1}\\ & \text{i}& & 1& & -1& & \text{i}\\ & \text{i}& & -1& & 1& & \text{i}\\ & -1& & \text{i}& & \text{i}& & 1\end{array}\right]\end{array}$$

mit: $${\text{U}}_{\text{ST}}=\frac{1}{\sqrt{4}}\left[\begin{array}{cc}1& \text{i}\\ \text{i}& 1\end{array}\right]\times \left[\begin{array}{cc}1& \text{i}\\ \text{i}& 1\end{array}\right]$$

The following notation is introduced for further discussion: flies a photon in the direction 1 (Path a and c ) the state | 1> _{i is} assigned to the photon i (i = 1,2). The photon i flies in that direction 0 (Path b and d ), the state | 0> _{i is} assigned to it. Before the two independent, identical photons use the beam splitter ST can reach the state $${\mathrm{\Psi }}_{}$$$${\mathrm{}}_1=\left|0{>}_{\mathrm{2nd}}\right|1{>}_1=|0.1>$$

be assigned. The state transformation brought about by the symmetrical beam splitter can then be carried out for two distinguishable photons (this can be achieved, for example, if the two identical photons are so temporally offset on the beam splitter ST that the wave trains (wave packets) assigned to the photons do not overlap at the beam splitter) by the unitary, local operator U _ST describe [5], [6]: $${\text{U}}_{\text{ST}}=\frac{1}{\sqrt{\mathrm{4th}}}\begin{array}{c}|0.0〉\\ |0.1〉\\ |1.0〉\\ |1.1〉\end{array}\begin{array}{c}\begin{array}{cccc}〈0.0|& 〈0.1|& 〈1.0|& 〈1.1|\end{array}\\ \left[\begin{array}{cccccccc}& 1& & \text{i}& & \text{i}& & \text{-1}\\ & \text{i}& & 1& & -1& & \text{i}\\ & \text{i}& & -1& & 1& & \text{i}\\ & -1& & \text{i}& & \text{i}& & 1\end{array}\right]\end{array}$$

With: $${\text{U}}_{\text{ST}}=\frac{1}{\sqrt{\mathrm{4th}}}\left[\begin{array}{cc}1& \text{i}\\ \text{i}& 1\end{array}\right]\times \left[\begin{array}{cc}1& \text{i}\\ \text{i}& 1\end{array}\right]$$

$${\mathrm{\Psi }}_2=1/4^2\left(\text{i}|1{>}_2|1{>}_1+\text{i}|0{>}_2|0{>}_1+|0{>}_2|1{>}_1-|1{>}_2|0{>}_1\right).$$

For the leaking state at the beam splitter Ψ ₂ you get with: $${\mathrm{\Psi }}_{\mathrm{2nd}}={\text{<figure-callout id="1" label="U ST Ψ" filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png" state="{{state}}">U </figure-callout>}}_{\text{ST}}{\mathrm{\Psi }}_{}{\mathrm{}}_1$$

$${\mathrm{\Psi }}_{\mathrm{2nd}}=1/{\mathrm{4th}}^{\mathrm{2nd}}\left(\text{i}|1{>}_{\mathrm{2nd}}|1{>}_1+\text{i}|0{>}_{\mathrm{2nd}}|0{>}_1+|0{>}_{\mathrm{2nd}}|1{>}_1-|1{>}_{\mathrm{2nd}}|0{>}_1\right).$$

• Die am weitesten verbreitete und experimentell gut bestätigte Vorstellung, zu der Frage, wann man von unterscheidbaren, identischen Quantensystemen sprechen kann, lässt sich am einfachsten anhand eines Beispiels erläutern: Hierzu bieten sich die in [7] beschriebenen Experimente an, bei denen einzelne oder auch mehrere ⁴⁰Ca⁺-Ionen in einer linearen Ionenfalle gespeichert werden können. Werden in der in [7] beschriebenen linearen Ionenfalle beispielsweise zwei ⁴⁰Ca⁺-Ionen gespeichert, so sind die einzelnen Ionen in einem, in guter Näherung etwa kugelförmigen Raumbereich mit einem Durchmesser von etwa 2 µm lokalisiert. Die beiden Raumbereiche R₁ (enthält das Ion 1) und R₂ (enthält das Ion 2) haben dabei einen Abstand von etwa 5 µm. Die während des Betriebs der linearen Ionenfalle einzuhaltenden Randbedingungen stellen dabei sicher, dass mit an Sicherheit grenzender Wahrscheinlichkeit ausgeschlossen werden kann, dass die beiden Ionen ihre Plätze tauschen. Genau dieses ist der entscheidende Punkt. Die beiden Ionen können genau dann als unterscheidbar angesehen werden, wenn diesen aufgrund der Randbedingungen klar unterscheidbare Raumbereiche so zugeordnet werden können, dass diese räumlich voneinander isoliert sind. Überlappen zu irgendeinem Zeitpunkt die beiden Raumbereiche R₁ und R₂ , so können die beiden Ionen ab diesem Zeitpunkt im Allgemeinen nicht mehr als unterscheidbar angesehen werden.

Since the question of when two physically completely identical quantum systems can be regarded as distinguishable repeatedly gives rise to controversial discussions, it makes sense to briefly consider the term "distinguishable, identical quantum systems" at this point:

• The most widespread and experimentally well-confirmed idea, when to speak of distinguishable, identical quantum systems, is easiest to explain using an example: The experiments described in [7], in which individual or also several ⁴⁰ Ca ⁺ ions can be stored in a linear ion trap. If, for example, two ⁴⁰ Ca ⁺ ions are stored in the linear ion trap described in [7], the individual ions are localized in a roughly spherical area with a diameter of approximately 2 µm. The two room areas R ₁ (contains the ion 1 ) and R ₂ (contains the ion 2nd ) have a distance of about 5 µm. The boundary conditions to be observed during the operation of the linear ion trap ensure that it is almost certain that the two ions can switch places. This is exactly the crucial point. The two ions can be regarded as distinguishable if and only if they can be assigned to clearly distinguishable spatial areas due to the boundary conditions so that they are spatially isolated from one another. At some point, the two areas overlap R ₁ and R ₂ , the two ions can generally no longer be regarded as distinguishable from this point in time.

This idea can now be applied to the in schematically represented situation in which two identical photons on the symmetrical beam splitter ST meet: To do this, of course, in a first step you have to assign spatial areas to the two photons that describe where the individual photons are located at a certain point in time. Since photons move at the speed of light, this must of course also apply to the corresponding areas. The smallest conceivable spatial areas are therefore spatial areas that move with the individual photons and correspond in the direction of propagation to the expansion of the wave packets assigned to the individual photons (which are described by the respective states) and are defined perpendicular to the direction of propagation by the spatial radiation characteristics of the respective source . It should also be noted that a single photon, after it has passed the beam splitter, must be assigned a superposition state. Behind the beam splitter you have to assign a corresponding area to each of the two state components. Furthermore, it is an experimentally well confirmed fact that the energy assigned to a photon is not split up at the beam splitter. The energy of a photon is therefore either completely reflected or completely transmitted at the beam splitter. After the beam splitter, the energy assigned to a photon can therefore only ever be assigned to exactly one of the two state components. In principle, it is impossible to predict which part of the state the energy of the photon must be assigned to.

The energy assigned to a photon is therefore always located in one of the “possible” spatial areas at that time. Either in the area assigned to the reflected portion of the state or in the area assigned to the transmitted portion of the state. In order to keep the use of language as simple as possible, we should also speak of the “possible” room areas for a single photon if the photon is still in front of the beam splitter and of course only one room area can be assigned to it.

For those in The situation is shown schematically: The two photons, the one from the source Q1 emitted photon 1 and that from the source Q2 emitted photon 2nd , can thus be regarded as distinguishable at a certain point in time if those for the photon 1 at this time possible spatial areas with those for the photon 2nd do not overlap possible room areas at this time. If the spatial areas overlap from a certain point in time, depending on the degree of overlap, the two photons can generally no longer be regarded as distinguishable from this point in time.

• Fall 1: Die den beiden Photonen zugeordneten Raumbereiche befinden sich noch vor dem Strahlteiler (siehe hierzu auch ): Da, die den beiden Photonen zu einem bestimmten Zeitpunkt T_A zugeordneten, Raumbereiche R₁ (Photon 1) und R₂ (Photon 2) aufgrund der Randbedingungen (der gewählten Anordnung) prinzipiell erst dann überlappen können, wenn diese den Strahlteiler erreichen, können die beiden Photonen vor dem Strahlteiler mit Sicherheit als unterscheidbar angesehen werden. Dabei spielt es keine Rolle, zu welchen Zeitpunkten die beiden Quellen die einzelnen Photonen emittieren.
• Fall 2: Die, den beiden Photonen zugeordneten, Raumbereiche erreichen den Strahlteiler zeitlich so versetzt, dass die für das Photon 1 möglichen Raumbereiche mit den für das Photon 2 möglichen Raumbereichen am Strahlteiler nicht überlappen: In diesem Fall können die beiden Photonen auch nach dem Strahlteiler mit Sicherheit als unterscheidbar angesehen werden, da die für das Photon 1 nach dem Strahlteiler möglichen Raumbereiche R_T1 und R_R1 mit den für das Photon 2 nach dem Strahlteiler möglichen Raumbereiche R_T2 und R_R2 aufgrund der Randbedingungen dann natürlich prinzipiell nicht überlappen können. Dabei bezeichnet R_Ti den Raumbereich, der dem transmittierten Zustandsanteil von Photon i (i = 1,2) zu einem bestimmten Zeitpunkt T_B zugeordnet wird und R_Ri den Raumbereich, der dem reflektierten Zustandsanteil von Photon i zu dem Zeitpunkt T_B zugeordnet wird (siehe hierzu auch ). Als auslaufender Zustand am Strahlteiler ergibt sich im Fall 2 daher gemäß Gl. (4) der Produktzustand Ψ₂ . Eine vollkommen andere Situation liegt vor, wenn beide Quellen zeitgleich je ein Photon emittieren.
• Fall 3: Beide Quellen emittieren zeitgleich je ein Photon: In diesem Fall treffen die, den einzelnen Photonen zugeordneten, Raumbereiche am Strahlteiler gleichzeitig ein. Ab dem Strahlteiler beginnen die für das Photon 1 möglichen Raumbereiche mit den für das Photon 2 möglichen Raumbereiche einander zu überlappen. Ab dem Zeitpunkt, ab dem die den beiden Photonen zugeordneten Raumbereiche den Strahlteiler vollständig passiert haben, überlappen die für das Photon 1 möglichen Raumbereiche maximal mit den für das Photon 2 möglichen Raumbereichen (R_R1 = R_T2 und R_T1 = R_R2). Die beiden Photonen müssen daher ab diesem Zeitpunkt als ununterscheidbar angesehen werden.

Based on this idea, the in differentiate between three cases:

• case 1 : The spatial areas assigned to the two photons are still in front of the beam splitter (see also ): There, the two photons at a certain time T _A assigned, room areas R ₁ (Photon 1 ) and R ₂ (Photon 2nd ) can only overlap due to the boundary conditions (of the chosen arrangement) when they reach the beam splitter, the two photons in front of the beam splitter can certainly be regarded as distinguishable. It does not matter at what times the two sources emit the individual photons.
• case 2nd : The spatial areas assigned to the two photons reach the beam splitter at different times so that those for the photon 1 possible spatial areas with those for the photon 2nd possible space areas on the beam splitter do not overlap: In this case, the two photons can certainly be regarded as distinguishable even after the beam splitter, since the photons 1 possible areas after the beam splitter R _T1 and R _R1 with those for the photon 2nd possible areas after the beam splitter R _T2 and R _R2 because of the boundary conditions, of course, they cannot overlap in principle. Inscribed R _Ti the spatial area that corresponds to the transmitted state portion of Photon i (i = 1,2) at a certain point in time T _B is assigned and R _Ri the spatial area that corresponds to the reflected state portion of Photon i at the time T _B is assigned (see also ). In the case, the resulting state at the beam splitter results 2nd therefore according to Eq. (4) the product condition Ψ ₂ . The situation is completely different if both sources emit a photon at the same time.
• case 3rd : Both sources emit one photon at a time: In this case, the spatial areas assigned to the individual photons arrive at the beam splitter at the same time. Starting from the beam splitter, those for the photon begin 1 possible spatial areas with those for the photon 2nd possible areas of space overlap each other. From the point in time at which the spatial areas assigned to the two photons have completely passed the beam splitter, those for the photon overlap 1 maximum possible room areas with those for the photon 2nd possible areas (R _R1 = R _T2 and R _T1 = R _R2 ). From this point on, the two photons must therefore be regarded as indistinguishable.

The case 3rd is the relevant case in this context. Since the two photons must be regarded as indistinguishable after the beam splitter in this case, the state can Ψ ₂ should not be regarded as a state at the beam splitter, since it does not have the symmetry properties required for indistinguishable quantum systems. According to the spin statistics theorem valid in quantum physics [8th] for indistinguishable photons (generally for bosons), the state ^B Ψ ₃ at the beam splitter emerges from the state Ψ ₂ by looking at the state Ψ ₂ symmetrized. The state ^B Ψ ₃ is then given by $${}^{\text{B}}{\mathrm{\Psi }}_{\mathrm{3rd}}=\text{i}/{\mathrm{2nd}}^{1/\mathrm{2nd}}\left(|1{>}_{\mathrm{2nd}}|1{>}_1+|0{>}_{\mathrm{2nd}}|0{>}_1\right).$$

ergeben. Den korrekten Ausdruck für den am Strahlteiler auslaufenden Zustand erhält man somit erst, nachdem man den Zustand Ψ₂ im Fall von Bosonen symmetrisiert oder im Fall von Fermionen anti-symmetrisiert hat. Erst die Anwendung des Spin-Statistik-Theorems auf den Zustand Ψ₂ führt somit zu einer korrekten Beschreibung der betrachteten Situation am Strahlteiler ST.If it were not bosons but two fermions, the condition would have to be Ψ ₂ still anti-symmetrize. In the case of fermions, the condition would change $${}^{\text{F}}{\mathrm{\Psi }}_{\mathrm{3rd}}=1/{\mathrm{<figure-callout\; id="1"\; label="2nd"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">2nd</figure-callout>}}^{1/\mathrm{2nd}}\left(|0{>}_{\mathrm{2nd}}|1{>}_1-|1{>}_{\mathrm{2nd}}|0{>}_1\right)$$

surrender. The correct expression for the state at the beam splitter can only be obtained after the state Ψ ₂ symmetrized in the case of bosons or anti-symmetrized in the case of fermions. Only the application of the spin statistics theorem to the state Ψ ₂ thus leads to a correct description of the situation under consideration at the beam splitter ST .

$$\text{mit:}|\text{A}>={\text{e}}^{-\text{i}\pi \text{/2}}{\mathrm{\Phi }}_{+}\text{ und }|\text{B}>={\mathrm{\Psi }}_{-},\text{ wobei gilt:}<\text{A}|\text{B}>=0.$$

If the previous considerations apply, then one can apply the spin statistics theorem to the state Ψ ₂ as a state transformation. The unitary operator that formally describes this state transformation can be determined simply by looking at the state Ψ ₂ circumscribes. Regarding the Bell base you get: $$\begin{array}{c}{\mathrm{\Psi }}_{\mathrm{2nd}}=1/{\mathrm{<figure-callout\; id="1"\; label="4th"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">4th</figure-callout>}}^{1/\mathrm{2nd}}\left(\text{i}|1{>}_{\mathrm{2nd}}|1{>}_1+\text{i}|0{>}_{\mathrm{2nd}}|0{>}_1+|0{>}_{\mathrm{2nd}}|1{>}_1-|1{>}_{\mathrm{2nd}}|0{>}_1\right)\\ =1/{\mathrm{<figure-callout\; id="1"\; label="2nd"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">2nd</figure-callout>}}^{1/\mathrm{2nd}}\left(\text{i}{\mathrm{\Phi }}_{+}-{\mathrm{\Psi }}_{-}\right)={\text{e}}^{\text{i}\pi }/{\mathrm{2nd}}^{1/\mathrm{2nd}}\left({\text{e}}^{-\text{i}\pi /\mathrm{2nd}}{\mathrm{\Phi }}_{+}+{\mathrm{\Psi }}_{-}\right)\\ ={\text{e}}^{\text{i}\pi }/{\mathrm{2nd}}^{1/\mathrm{2nd}}\left(|\text{A}>+|\text{B}>\right).\end{array}$$

$$\text{With:}|\text{A}>={\text{e}}^{-\text{i}\pi \text{/ 2nd}}{\mathrm{\Phi }}_{+}\text{and}|\text{B}>={\mathrm{\Psi }}_{-},\text{where:}<\text{A}|\text{B}>=0.$$

so erhält man: $${\text{U}}_{\text{NLB}}{\mathrm{\Psi }}_2={\text{e}}^{\text{i}\pi }|\text{A}>=\text{i}{\mathrm{\Phi }}_{+}={}^{\text{B}}\mathrm{\Psi }{}_3$$

The state Ψ ₂ can thus also be formally understood as an element of a two-dimensional vector space that is spanned by the orthogonal state vectors | A> and | B>. If one defines the unitary operator on this vector space U _NLB according to $${\text{U}}_{\text{NLB}}=\frac{1}{\sqrt{\mathrm{2nd}}}\begin{array}{c}|\text{A}〉\\ |\text{B}〉\end{array}\begin{array}{c}\begin{array}{cc}〈\text{A}|& 〈\text{B}|\end{array}\\ \left[\begin{array}{cc}1& 1\\ -1& 1\end{array}\right]\end{array}$$

how to get: $${\text{U}}_{\text{NLB}}{\mathrm{\Psi }}_{\mathrm{2nd}}={\text{e}}^{\text{i}\pi }|\text{A}>=\text{i}{\mathrm{\Phi }}_{+}={}^{\text{B}}\mathrm{\Psi }{}_{\mathrm{3rd}}$$

$${\text{mit: U}}_{\text{NLF}}={\left({\text{U}}_{\text{NLB}}\right)}^{-1}.$$

For the considered situation at the beam splitter, the effect of the application of the spin statistics theorem on the state can be Ψ ₂ for bosons formally using the operator U _NLB to be discribed. In the event that the two quantum systems are not bosons but fermions, you immediately get: $${\text{U}}_{\text{NLF}}{\mathrm{\Psi }}_{\mathrm{2nd}}={\text{e}}^{\text{i}\pi }|\text{B}>=-{\mathrm{\Psi }}_{-}={}^{\text{F}}\mathrm{\Psi }{}_{\mathrm{3rd}}$$

$${\text{with U}}_{\text{NLF}}={\left({\text{U}}_{\text{NLB}}\right)}^{-1}.$$

und für den Fall von zwei identischen Fermionen die Gleichung $${}^{\text{F}}\mathrm{\Psi }{}_3={\text{U}}_{\text{NLF}}{\text{U}}_{\text{ST}}|\mathrm{0,1}>.$$

U _NLF is thereby through to U _NLB given inverse operator. Overall, the equation results for the state transformation which converts the incoming state into the outgoing state in the case of two identical bosons $${}^{\text{B}}\mathrm{\Psi }{}_{\mathrm{3rd}}={\text{U}}_{\text{NLB}}{\text{U}}_{\text{ST}}|0.1>$$

and in the case of two identical fermions, the equation $${}^{\text{F}}\mathrm{\Psi }{}_{\mathrm{3rd}}={\text{U}}_{\text{NLF}}{\text{U}}_{\text{ST}}|0.1>.$$

From the fact that the operators U _NLB and U _NLF converting a product state into an entangled state, it follows that these must be non-local operators. The crucial question now is the following: Can the operators U _NLB and U _NLF to be understood as interaction operators? Since the question of whether every physically feasible, non-local, unitary operator describing a state transformation cannot be understood as an interaction operator, it has so far not been possible to answer the general question within the framework of the formalism on which quantum physics is based, so there is only the possibility of considering each individual case separately . For the example discussed above, when two identical photons hit the symmetrical beam splitter at the same time, it can easily be shown that the operator U _NLB cannot be regarded as an interaction operator: so that the operator U _NLB as an interaction operator, one must also be able to assign an interaction to it. However, since it has so far not been possible to specify an interaction via which the two photons on the beam splitter can interact with one another, the operator can U _NLB in this example also not to be understood as an interaction operator.

As this example shows, there are physically realizable, non-local, unitary operators describing a state transformation that cannot be regarded as interaction operators.

That the approach described above is in describes the schematically represented situation correctly, you can also see it when you see it in the picture 2nd . Quantization describes. On the in schematically represented situation is in [ 9 ] in the chapter 6.8 (Eq. (6.8.6) to (6.8.20)) in detail in the image of 2nd . Quantization received. The equations given there are generally valid. These not only allow a description of the borderline cases discussed above (case 2nd and case 3rd ), but also enable a description in the event that the wave packets assigned to the photons partially overlap at the beam splitter.

beschrieben werden. Nach den Überlegungen in [9] können die beiden Photonen in der oben betrachteten Situation, solange diese sich noch vor dem Strahlteiler ST befinden, als voneinander unabhängig angesehen werden, da für die Quellen Q1 und Q2 angenommen wurde, dass diese unabhängig voneinander betrieben werden können. Ein Indiz dafür, dass diese Annahme zulässig ist, liefern beispielsweise auch die in [10] beschriebenen Experimente.In the event that one is only interested in the two borderline cases discussed above, the in [ 9 ] simplify the given equations. For this purpose, the following in [ 9 ] symbols and definitions introduced and the results briefly reproduced. For details, see [ 9 ] referred. If the two photons can be regarded as independent of each other, the one on the symmetrical, lossless beam splitter ST incoming state then by the state $$\begin{array}{c}|\text{IN}>={\displaystyle \int \text{German}{\displaystyle \int \text{dt '}\mathrm{\xi }\left(\text{t}\right)\mathrm{\xi }\left(\text{t '}\right){\stackrel{⌢}{\text{a}}}^{+}{}_{\text{1}}\left(\text{t}\right)}}{}^{\stackrel{⌢}{\text{a}}}{}_{\text{2nd}}\left(\text{t '}\right)|0>=\left({\displaystyle \int \text{German}}\mathrm{}\mathrm{\xi }\left(\text{t '}\right){\stackrel{⌢}{\text{a}}}^{+}{}_{\text{1}}\left(\text{t}\right)\right)\left({\displaystyle \int \text{dt '}}\mathrm{\xi }\left(\text{t '}\right){\stackrel{⌢}{\text{a}}}^{+}{}_{\text{2nd}}\left(\text{t '}\right)\right)|0>\\ ={\stackrel{⌢}{\text{a}}}^{+}{}_{\text{1}\text{,}\mathrm{\xi }\mathrm{,}\left(\text{t}\right)}{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{2nd}\text{,}\mathrm{\xi }\mathrm{,}\left(\text{t '}\right)}|0>=|1{>}_{\mathrm{1,}\mathrm{\xi }\left(\text{t}\right)}|1{>}_{\mathrm{2,}\mathrm{\xi }\left(\text{t '}\right)}\end{array}$$

to be discribed. After the considerations in [ 9 ] the two photons can in the situation considered above, as long as they are still in front of the beam splitter ST are considered to be independent of each other because of the sources Q1 and Q2 it was assumed that these can be operated independently of one another. An indication that this assumption is permissible is provided, for example, by those in [ 10 experiments described.

beschrieben werden. Überlappen die den Photonen zugeordneten Wellenpakete am Strahlteiler nicht, so erhält man den am Strahlteiler auslaufenden Zustand gemäß $$\begin{array}{c}{|\text{OUT}}_2>=\text{i/2}\left(|1{>}_{\mathrm{3,}\mathrm{\xi }\left(\text{t}\right)}|1{>}_{\mathrm{3,}\mathrm{\xi }\left(\text{t'}\right)}+|1{>}_{\mathrm{4,}\mathrm{\xi }\left(\text{t}\right)}|1{>}_{\mathrm{4,}\mathrm{\xi }\left(\text{t'}\right)}\right)\\ +1/2\left(|1{>}_{\mathrm{4,}\mathrm{\xi }\left(\text{t}\right)}|1{>}_{\mathrm{3,}\mathrm{\xi }\left(\text{t'}\right)}-|1{>}_{\mathrm{3,}\mathrm{\xi }\left(\text{t}\right)}|1{>}_{\mathrm{4,}\mathrm{\xi }\left(\text{t'}\right)}\right).\end{array}$$

After the two photons have passed the beam splitter, the quantum system can for the limit cases considered here by the state $$\begin{array}{c}|\text{OUT}>={\displaystyle \int \text{German}}{\displaystyle \int \text{dt '}\mathrm{\xi }\left(\text{t}\right)\mathrm{\xi }\left(\text{t '}\right)[\mathrm{2nd}\left({\stackrel{⌢}{\text{a}}}^{+}{}_{\mathrm{3rd}}\left(\text{t}\right){\stackrel{⌢}{\text{a}}}^{+}{}_{\mathrm{3rd}}\left(\text{t '}\right)+{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{4th}}\left(\text{t}\right){\stackrel{⌢}{\text{a}}}^{+}{}_{\text{4th}}\left(\text{t '}\right)\right)}\\ +1/\mathrm{2nd}\left({\stackrel{⌢}{\text{a}}}^{+}{}_{\text{4th}}\left(\text{t}\right){\stackrel{⌢}{\text{a}}}^{+}{}_{\mathrm{3rd}}\left(\text{t '}\right)-{\stackrel{⌢}{\text{a}}}^{+}{}_{\mathrm{3rd}}\left(\text{t}\right){\stackrel{⌢}{\text{a}}}^{+}{}_{\text{4th}}\left(\text{t '}\right)\right)]|0>\\ =[\text{i}/\text{2nd}\left({\stackrel{⌢}{\text{a}}}^{+}{}_{\mathrm{3,}\mathrm{\xi }\left(\text{t}\right)}{\stackrel{⌢}{\text{a}}}^{+}{}_{\mathrm{3,}\mathrm{\xi }\left(\text{t '}\right)}+{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{4th}\text{,}\mathrm{\xi }\left(\text{t}\right)}{\stackrel{⌢}{\text{a}}}^{+}{}_{\mathrm{3,}\mathrm{\xi }\left(\text{t '}\right)}\right)\\ +1/\mathrm{2nd}\left({\stackrel{⌢}{\text{a}}}^{+}{}_{\text{4th}\text{,}\mathrm{\xi }\left(\text{t}\right)}{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{3rd}\text{,}\mathrm{\xi }\left(\text{t '}\right)}-{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{3rd}\text{,}\mathrm{\xi }\left(\text{t}\right)}{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{4th}\text{,}\mathrm{\xi }\left(\text{t '}\right)}\right)]|0>\end{array}$$

to be discribed. If the wave packets assigned to the photons do not overlap on the beam splitter, the state which is running out on the beam splitter is obtained accordingly $$\begin{array}{c}{|\text{OUT}}_{\mathrm{2nd}}>=\text{i / 2}\left(|1{>}_{\mathrm{3,}\mathrm{\xi }\left(\text{t}\right)}|1{>}_{\mathrm{3,}\mathrm{\xi }\left(\text{t '}\right)}+|1{>}_{\mathrm{4,}\mathrm{\xi }\left(\text{t}\right)}|1{>}_{\mathrm{4,}\mathrm{\xi }\left(\text{t '}\right)}\right)\\ +1/\mathrm{2nd}\left(|1{>}_{\mathrm{4,}\mathrm{\xi }\left(\text{t}\right)}|1{>}_{\mathrm{3,}\mathrm{\xi }\left(\text{t '}\right)}-|1{>}_{\mathrm{3,}\mathrm{\xi }\left(\text{t}\right)}|1{>}_{\mathrm{4,}\mathrm{\xi }\left(\text{t '}\right)}\right).\end{array}$$

The state transformation (beam splitter property) assigned to the transition of the incoming state | IN> into the outgoing state | OUT ₂ > then corresponds exactly to that in the image of the 1st quantization by the operator U _ST (Eq. (2)) described state transformation. As the considerations in [ 9 ] show, in this case the photons can be regarded as distinguishable even after the beam splitter.

umschreiben zu $$\begin{array}{c}|{\text{OUT}}_3>=\text{i}/2^{1/2}\left(|2{>}_{\mathrm{3,}\mathrm{\xi }}+|2{>}_{\mathrm{4,}\mathrm{\xi }}\right)\\ +1/2\left({\stackrel{⌢}{\text{a}}}^{+}{}_{\text{ 4}\text{,}\mathrm{\xi }\left(\text{t}\right)}{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{ 3}\text{,}\mathrm{\xi }\left(\text{t'}\right)}-{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{ 3}\text{,}\mathrm{\xi }\left(\text{t}\right)}{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{ 4}\text{,}\mathrm{\xi }\left(\text{t'}\right)}\right)|0>.\end{array}$$

In the event that the two photons reach the beam splitter at the same time, Eq. (SQ.2) using the equation $$|\mathrm{2nd}{>}_{\text{i},\mathrm{\xi }}=1/{\mathrm{<figure-callout\; id="1"\; label="2nd"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">2nd</figure-callout>}}^{1/\mathrm{2nd}}{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{1}\text{,}\mathrm{\xi }}{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{i}\text{,}\mathrm{\xi }}|0>,\text{with i}=3.4$$

rewrite to $$\begin{array}{c}|{\text{OUT}}_{\mathrm{3rd}}>=\text{i}/{\mathrm{2nd}}^{1/\mathrm{2nd}}\left(|\mathrm{2nd}{>}_{\mathrm{3,}\mathrm{\xi }}+|\mathrm{2nd}{>}_{\mathrm{4,}\mathrm{\xi }}\right)\\ +1/\mathrm{2nd}\left({\stackrel{⌢}{\text{a}}}^{+}{}_{\text{4th}\text{,}\mathrm{\xi }\left(\text{t}\right)}{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{3rd}\text{,}\mathrm{\xi }\left(\text{t '}\right)}-{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{3rd}\text{,}\mathrm{\xi }\left(\text{t}\right)}{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{4th}\text{,}\mathrm{\xi }\left(\text{t '}\right)}\right)|0>.\end{array}$$

so erhält man für den am Strahlteiler auslaufenden Zustand den Ausdruck $$|{\text{OUT}}_3>={\text{i/2}}^{1/2}\left(|2{>}_{\mathrm{3,}\mathrm{\xi }}+|2{>}_{\mathrm{4,}\mathrm{\xi }}\right).$$

Now take into account that in this case applies $$\left({\stackrel{⌢}{\text{a}}}^{+}{}_{\text{4th}\text{,}\mathrm{\xi }\left(\text{t}\right)}{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{3rd}\text{,}\mathrm{\xi }\left(\text{t '}\right)}-{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{3rd}\text{,}\mathrm{\xi }\left(\text{t}\right)}{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{4th}\text{,}\mathrm{\xi }\left(\text{t '}\right)}\right)=\left[{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{4th}\text{,}\mathrm{\xi }\left(\text{t}\right)},{\stackrel{⌢}{\text{a}}}^{+}{}_{\text{3rd}\text{,}\mathrm{\xi }\left(\text{t '}\right)}\right]$$

in this way you get the printout for the state at the beam splitter $$|{\text{OUT}}_{\mathrm{3rd}}>={\text{i / 2}}^{1/\mathrm{2nd}}\left(|\mathrm{2nd}{>}_{\mathrm{3,}\mathrm{\xi }}+|\mathrm{2nd}{>}_{\mathrm{4,}\mathrm{\xi }}\right).$$

The state | OUT ₃ > thus corresponds exactly to the state ^B Ψ ₃ in the image of the first quantization. If we consider the transition from Eq. (SQ.2) to Eq. (SQ.7) as a state transformation, it can be seen that the effect of the commutator relation Eq. (SQ.6) consists of the state components in Eq., Which are only possible for distinguishable quantum systems. (SQ.2) to suppress. The application of the commutator relaton in connection with Eq. (SQ.4) to the state | OUT>, thus corresponds exactly to the application of the operator U _NLB on the condition Ψ ₂ in the picture of the 1st quantization.

It follows from this that the two identical photons, if they arrive at the beam splitter at the same time, in the image of 2nd . Quantization after the beam splitter must be regarded as indistinguishable.

As these considerations show, the description in the picture corresponds to 2nd . Quantization thus the description in the image of the 1st quantization. However, it was also not recognized in [9] that, as this example shows, there are physically realizable, non-local, unitary operators describing a state transformation that cannot be regarded as interaction operators.

• Der Operator U_STQ lässt sich mit Hilfe der Pauli-Matrix σ₂ schreiben zu: $${\text{U}}_{\text{STQ}}=\frac{1}{\sqrt{2}}\left[\left[\begin{array}{cc}0& -\text{i}\\ \text{i}& 0\end{array}\right]\times 1+1\times \left[\begin{array}{cc}0& \text{i}\\ -\text{i}& 0\end{array}\right]\right]$$
  
  $${\text{U}}_{\text{STQ}}=\frac{1}{\sqrt{2}}\begin{array}{c}|\mathrm{0,0}\rangle \\ |\mathrm{0,1}\rangle \\ |\mathrm{1,0}\rangle \\ |\mathrm{1,1}\rangle \end{array}\begin{array}{c}\begin{array}{cccc}\langle \mathrm{0,0}|& \langle \mathrm{0,1}|& \langle \mathrm{1,0}|& \langle \mathrm{1,1}|\end{array}\\ \left[\begin{array}{cccccccc}& \text{0}& & \text{i}& & \text{-i}& & \text{0}\\ & \text{-<figure-callout id="0" label="i" filenames="DE102019008600A1\_0063.png" state="{{state}}">i</figure-callout>}& & \text{0}& & \text{0}& & \text{-<figure-callout id="0" label="i i" filenames="DE102019008600A1\_0063.png" state="{{state}}">i</figure-callout>}& \\ & \text{i}& & \text{0}& & \text{0}& & \text{i}\\ & \text{0}& & \text{i}& & \text{-i}& & \text{0}\end{array}\right]\end{array}$$
  
  mit:
• $${\mathrm{<figure-callout\; id="2"\; label="\sigma "\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">\sigma </figure-callout>}}_2=\left[\begin{array}{cc}0& -\text{i}\\ \text{i}& 0\end{array}\right]$$
  

When two identical, distinguishable photons hit a symmetrical beam splitter at the same time (case 3rd ), the operator U _STQ = U _NLB U _ST , which describes the state transformation at the beam splitter, has a remarkable property:

• The operator U _STQ can be written using the Pauli matrix σ ₂ : $${\text{U}}_{\text{STQ}}=\frac{1}{\sqrt{\mathrm{2nd}}}\left[\left[\begin{array}{cc}0& -\text{i}\\ \text{i}& 0\end{array}\right]\times 1+1\times \left[\begin{array}{cc}0& \text{i}\\ -\text{i}& 0\end{array}\right]\right]$$
  
  $${\text{U}}_{\text{STQ}}=\frac{1}{\sqrt{\mathrm{2nd}}}\begin{array}{c}|0.0〉\\ |0.1〉\\ |1.0〉\\ |1.1〉\end{array}\begin{array}{c}\begin{array}{cccc}〈0.0|& 〈0.1|& 〈1.0|& 〈1.1|\end{array}\\ \left[\begin{array}{cccccccc}& \text{0}& & \text{i}& & \text{-i}& & \text{0}\\ & \text{-i}& & \text{0}& & \text{0}& & \text{-i}\\ & \text{i}& & \text{0}& & \text{0}& & \text{i}\\ & \text{0}& & \text{i}& & \text{-i}& & \text{0}\end{array}\right]\end{array}$$
  
  With:
• $${\mathrm{\sigma }}_{\mathrm{2nd}}=\left[\begin{array}{cc}0& -\text{i}\\ \text{i}& 0\end{array}\right]$$
  

The operator U _STQ now has the following remarkable property: In a first step (due to the effect of U _ST ) the energetically represented quantum system is in the state | 0.1> in the elementary quantum system in the state Ψ ₂ transferred. In a second step (due to the effect of U _NLB ) the elementary quantum system becomes in the state Ψ ₂ transferred to an energetically represented quantum system in state ^B Ψ ₃ (see also [ 2nd ] and the notes below).

Elementary quantum states must be assigned to the internal properties that physicist and philosopher David Bohm postulates. As folded properties of the implicit order (see also [ 14 ]). The effect of the Hermitian operator U _STQ is therefore primarily within the implicit order (due to the non-local effect of U _NLB ) and thus only has an indirect effect on the physical world (on energetically represented quantum systems).

• Postulat PIO: Hermitsche und unitäre Operatoren die als primär elementare Operatoren aufgefasst werden können, wirken primär innerhalb der impliziten Ordnung. Erst durch den Vorgang des Entfaltens wird der Wirkung dieser Operatoren in der physikalischen Welt eine energetische Repräsentation zugeordnet und diese damit sichtbar (messbar).

The effect of the in [ 2nd ] on page 23 specified unitary operator U _KV lies primarily within the implicit order. This describes the process of coherently coupled vacuum fluctuations, in which an elementary quantum system is converted into an energetic one. The operator too U _KV acts like U _NLB elementary states. In the following, non-local operators, which act on elementary states and convert an elementary quantum system into an energetically represented one, are referred to as primary elementary operators. This term is also used when the operator first transforms an energetically represented quantum system into an elementary one (like the operator U _STQ ) transferred and only in a second step an elementary quantum system into an energetic one. Therefore it is postulated here:

• Postulate PIO: Hermitian and unitary operators, which can be regarded as primary elementary operators, operate primarily within the implicit order. It is only through the process of unfolding that the effect of these operators in the physical world is assigned an energetic representation and is therefore visible (measurable).

This has far-reaching consequences: effects within the implicit order are fundamentally non-local and have an immediate effect [ 14 ]. However, these effects only seem to have an impact on the degree of spin freedom and on general conservation laws, such as energy or momentum conservation. However, it is still unclear why this is so.

There are also concrete arguments for consciousness to be assigned very fundamentally and thus in particular human consciousness and the abilities ascribed to it, such as that of perception combined with sensations and feelings, to the implicit order (see [ 14 ] and [ 2nd ]). In contrast to the approach of David Bohm, alternative approaches, such as the materialistic approach advocated by the vast majority of consciousness researchers that consciousness arises as an emergent property in sufficiently complex systems (brains), still have to show how this should be done . The statement "somehow" is not sufficient. Only: There is nothing more to be presented in the context of these approaches.

• Postulat PIB: Bewusstsein und die diesem zugeordneten Eigenschaften sind Teil der impliziten Ordnung. Die Interaktion von Bewusstsein mit der physikalischen Welt (der Welt der Materie und Energie) kann mittels primär elementarer Operatoren formal beschrieben werden. Diese Interaktion wird durch den Vorgang des Entfaltens bewirkt und damit sichtbar (messbar). Ein möglicher Prozess für eine Interaktion von Bewusstsein und Gehirn ist der in [2] beschriebene Prozess der kohärent gekoppelten Vakuumfluktuationen.

The following is postulated for consciousness and the properties associated with it:

• Postulate PIB: consciousness and the properties assigned to it are part of the implicit order. The interaction of consciousness with the physical world (the world of matter and energy) can be formally described using primarily elementary operators. This interaction is brought about by the unfolding process and is therefore visible (measurable). A possible process for an interaction between consciousness and brain is that in [ 2nd ] described process of coherently coupled vacuum fluctuations.

Although certainly not every primary elementary operator describes an interaction of the brain with our consciousness or mind and it is not yet clear how the operators in question can be classified, primary elementary operators offer the possibility of the interaction of consciousness or mind for the first time to be able to research the brain in the natural sciences using an established formalism.

The argument is often made that quantum processes (and the possible entanglements) can occur in the brain, but do not play a role in how information processing processes work, since the temperature in the brain is much too high for this and therefore coherent processes Quantum level can have no influence on the functioning of the brain and consciousness. This probably applies to energetically represented quantum systems. This argument certainly does not apply to elementary quantum systems. Elementary quantum systems lie outside the physical world. Decoherence due to material or energetic influences is therefore fundamentally excluded.

In order to better understand the basic idea of the method for interaction-free information transfer by means of photons, it is helpful to briefly consider the method for "interaction-free quantum measurement" described in the literature [ 11 ] (see also [ 12th ]). The basic structure of the in [ 11 ] proposed experiment is in shown schematically. A Mach-Zehnder interferometer is used for the “interaction-free measurement”. The Mach-Zehnder interferometer consists of the beam splitter ST1 , the two (ideal, lossless) mirrors SP1 and SP2 and the beam splitter ST2 . The experiment should show that it is possible to detect the presence of an absorber in one of the beam paths without interaction (without an exchange of energy). But is that true? To anticipate the answer: No.

In a strict sense, one cannot speak of a method for interaction-free measurement. To be able to measure something, at least two different measurement results must be possible. So that this method can be described as interaction-free, there must never be an energy exchange (an interaction) between the location where the object to be measured can be inserted into the beam path and the detectors used for the measurements. This condition is in [ 11 ] not met for all possible measurement results. No interaction is required for the measurement result, which clearly indicates the presence of the object in the beam path. But even for this measurement result, it is imperative that an energy exchange is fundamentally possible (and actually takes place in 50% of the cases if both beam paths are available), since otherwise the uniqueness caused by the method (the Structure) is ensured, is lost. Thus, in a strict sense, one cannot speak of a method for interaction-free measurement. The experiment only confirms the existence of non-local effects, as already known from the double slit experiment.

This is easy to see. Assuming that an experimenter "Alice" can freely decide when to use an absorber (AB) at the point ( A ) into the beam path, a second experimenter "Bob" would have to use the device available to him (consisting of the symmetrical, lossless beam splitter ST2 and the two detectors D1 and D2 ) can recognize when Alice is using your absorber ( FROM ) has spent in the beam path and when not. For this purpose, however, energy from Alice's room area must not be transferred to Bob's room area at any time, since otherwise it cannot be a method for interaction-free measurement.

If photons would behave like classic particles, the detectors can D1 and D2 the presence of the absorber ( FROM ) can only be recognized by halving the sum of the number of photons detected by both detectors per time. On average, for example, without an absorber 1000 Detected photons per second, with absorbers it is only 500 photons per second. It was assumed that the mean of the source Q emitted number of photons per second remains constant.

• „Der Zustand eines Quantensystems ist dem durchlaufenen speziellen Präparationsverfahren zugeordnet. Unter einem Quantenzustand (quantum state) verstehen wir dasjenige mathematische (!) Objekt, das es erlaubt, eindeutig die Wahrscheinlichkeit für die Ergebnisse aller möglichen Messungen an Systemen zu berechnen, die das zugeordnete Präparationsverfahren durchlaufen haben. Der Quantenzustand charakterisiert somit das Präparationsverfahren.“

How is the experiment described in the context of quantum physics? Formally, the standard interpretation of quantum physics is used (see, for example, [ 13 ]). The central element for the description of an experiment is the state. A state is then understood to mean the following (see [ 13 ], Page 30th ):

• “The state of a quantum system is assigned to the special preparation process that was carried out. We understand a quantum state to be the mathematical (!) Object that allows the probability of the results of all possible measurements to be unambiguously calculated on systems that have undergone the assigned preparation process. The quantum state thus characterizes the preparation process. "

In order to be able to describe an experiment, one must state for this one state. How do you get to the corresponding state? Strictly speaking, one is dependent on metaphors, on images. One transfers ideas from classical physics to quantum physics. And then comes statements like "on the one hand a photon shows particle properties, ... on the other hand it behaves like a wave" (see for example [ 12th ]). You assign properties to a single quantum system that are impossible in the context of classical physics and then pretend that you cannot assign a state to individual quantum systems (here photons) (the state formally describes the experiment, the situation under consideration , not a single photon!). Although in this way one obtains a state that adequately describes the possible results of the experiment. Which is astonishing in itself. The reason for these intellectual pull-ups can only be seen in the fact that it has so far not been possible to provide an adequate interpretation for quantum physics [ 14 ].

It is all the more astonishing that alternative approaches that avoid these difficulties are not considered relevant by the great majority of physicists. But why is that?

The physicist and philosopher David Bohm is together with the physicist F. David Peat in [ 15 ] investigated this question in detail. Bohm assumes that our internalized beliefs restrict our thinking, our perception and our actions. If we are not aware of this, we run the risk of seeing our ideas about the world as a good description of reality. In truth, however, we can only ever find the abstracted aspects of reality in the observed phenomena within the framework of our ideas. Don't describe reality itself. If one is not aware of these processes, the ideas easily become beliefs, they become a paradigm. Our beliefs then appear to us as reality. It is then no longer possible to work creatively with alternative ideas. Our intuition is blocked.

If one assumes that these ideas are correct, the question arises as to which beliefs are responsible for making it so difficult for quantum physics, alternative approaches such as the De Broglie-Bohm theory (Bohmian theory). Mechanics) [ 14 ] to discuss.

The vast majority of physicists have internalized the idea that the physical world (the entirety of matter and the interactions between it) is causally closed as a belief. This is not questioned. Indications that indicate that this cannot be the case are considered irrelevant and are ignored.

Bohm, on the other hand, assumes that what we call the physical world is only the "unfolded" part of an "implicit order" [ 14 ]. The structures and relationships within the physical world seem to be causally closed in the context of classical physics, since a cause can be given for each effect within the physical world. However, this is no longer the case in quantum physics. Already 1935 Einstein, Podolsky and Rosen (EPR) were the first to point out that quantum systems can interact with one another instantaneously over any distance and that in quantum physics neither a cause nor a plausible explanation can be given for these effects [ 16 ]. Bohm sees this as an indication that the physical world cannot be causally closed.

• „Unser Vorschlag, von der impliziten Ordnung als Grundlage auszugehen, besagt demnach, dass das, was primär, unabhängig existent und universell ist, vom Standpunkt der impliziten Ordnung aus ausgedrückt werden muss. Wir behaupten also, dass es die implizite Ordnung ist, die autonom wirkt, während die explizite Ordnung, wie zuvor dargelegt, aus einer Gesetzmäßigkeit der impliziten Ordnung resultiert, so dass sie sekundär, abgeleitet und nur in gewissen Zusammenhängen angebracht ist. Oder um es anders auszudrücken: Die Beziehungen, die das Grundgesetz bilden, bestehen zwischen eingefalteten Strukturen, die sich im Ganzen des Raums ineinander verweben und sich gegenseitig durchdringen, und nicht zwischen den abstrahierten und abgetrennten Formen, die sich den Sinnen (und auf unseren Instrumenten) manifestieren.“

Bohm therefore suggests that the description of the phenomena observable in the physical world no longer be based on the traditional mechanical order but on an implicit order (see [ 14 ], Page 241 ):

• “Our suggestion to use the implicit order as a basis therefore says that what is primary, independently existent and universal must be expressed from the point of view of the implicit order. We therefore claim that it is the implicit order that acts autonomously, while the explicit order, as stated above, results from a lawfulness of the implicit order, so that it is secondary, derived and only appropriate in certain contexts. Or to put it another way: the relationships that make up the Basic Law exist between folded structures that interweave and interpenetrate each other in the whole of the room, and not between the abstracted and separated forms that are available to the senses (and on our instruments ) manifest. "

Bohm defines an explicit order as relationships, structures that are evident in the observable phenomena, particularly the energetic properties, of the physical world. Bohm also regards the physical world as a developed part of the implicit order. Bohm's idea of “folding in” and “unfolding” is easiest to illustrate using an example: Bohm assumes that the “movement” of a “particle” in space takes place discontinuously (similar to the energetic transitions of an electron in an atom due to of the discrete energy states can only take place discontinuously). Bohm sees the movement of a particle as the constant unfolding and unfolding of an aspect of the implicit order ascribed to the particle. The properties attributed to the particle in the physical world, such as mass and charge, are then only the unfolded part of the implicit order. If the particle moves, the properties (folding) ascribed to the particle in the physical world disappear in order to reappear in another place (unfolding). Since this unfolding and unfolding is extremely fast due to the size of Plank's quantum of action h, these processes appear as a continuous movement.

Although Bohm's idea of an implicit order underlies the De Broglie-Bohm theory, this idea is not the subject of current basic research. The main reason for this can probably only be seen in the fact that it has so far not been possible to find experimental indications that distinguish the De Broglie-Bohm theory from alternative approaches, such as the standard interpretation of quantum physics.

In order to be able to show beyond any doubt that Bohm's idea of an implicit order is correct, one would have to find an experiment that allows the process of folding in and unfolding to be controlled by means of classically accessible quantities in such a way that the non-local nature of the implicit order is clear becomes recognizable. An experiment that could show this is in [ 1 ] first proposed and in [ 17th ] and [ 2nd ] in a simplified variant discussed experiment for the realization of an interaction-free information transfer. A successful implementation would be a strong indication that the idea of an implicit order is correct and that the physical world cannot be completed causally.

To do that in [ 1 ] To be able to adequately describe proposed methods for the interaction-free transfer of information was described in [ 1 ] presented a new interpretation approach for quantum physics. This was in [ 17th ] further developed and in [ 2nd ] with an adequate definition of entropy for the pure states considered there. Like Bohm, you go into [ 1 ] assume that the physical world cannot be completed causally. Therefore, intangible entities (in [ 1 ] referred to as internal properties) that act on the physical world without being part of the physical world. Which aspects of these internal properties can be effective in the physical world depends on the respective situation (the experimental setup chosen in each case). The effect of the internal properties on the physical world is described by means of the state that can be assigned to the quantum systems under consideration in the respective situation. The implicit order proposed by Bohm is much more general in approach, since it assumes that the physical world is an aspect of the implicit order. With regard to the basic ideas, as an intangible entity, the in 1 ] internal properties already included in the implicit order approach. New at the in [ 1 ] approach is that for the first time this opens up the possibility to experimentally test the idea introduced by Bohm of the unfolding and unfolding of the implicit order. Essential for this are the [ 1 ] terms introduced for the first time "elementary" quantum system and "energetically represented" quantum system. These terms refer to the relevant aspect of the quantum systems under consideration. In terms of spatial freedom, the quantum systems are represented energetically by the energy that is permanently assigned to them. For "particles" with a rest mass through the rest mass assigned to the particle and with photons through the energy assigned to the photons via the Plank quantum of action and the frequency. Except for brief moments, for example when a quantum system passes a beam splitter, particles have an energetic representation in the spatial degree of freedom (they are actually defined by them). In the degree of spin freedom, the quantum systems under consideration, as in [ 1 ] is shown in detail, both be represented energetically, but also exist as an elementary quantum system. In field-free space, these exist as elementary quantum systems with regard to the degree of spin freedom. If, on the other hand, a magnetic field is superimposed on the quantum systems, this process leads to an energetically represented quantum system in the spin degree of freedom. The energetic representation is then given by the interaction energy W = µ B, with with µ the magnetic moment and with B the superimposed magnetic field is referred to (see [1], Chapter IV). Spin ½ systems are considered as examples in [1]. If, for example, a magnetic field is superimposed on it in the z direction, the quantum system takes in the degree of spin freedom, depending on how the spin relates to the magnetic field B _z aligns the predetermined quantization axis, either the energy ΔE _z = µB _z or outputs it. The energy ΔE _z is therefore a clear measure for the alignment of the spin with respect to the quantization axis and can therefore be regarded as an energetic representation of the spin degree of freedom. If the quantum system is in a field-free space, the spin property is folded. If a magnetic field is superimposed on the quantum system, the property spin develops.

The process of unfolding and folding can thus be realized by switching a magnetic field on and off, which can be superimposed on the quantum system. Since magnetic fields are classically controllable quantities, they offer the possibility to experimentally test the process of folding in and unfolding that Bohm postulates using the experiment mentioned above to implement an interaction-free transfer of information and thereby to provide evidence that the implicit order approach is one provides an adequate description of the observable non-local phenomena and thus distinguishes them from other approaches.

If one considers the method for interaction-free quantum measurement proposed in [11], the following situation arises (see also ): Emits the source Q a photon, the quantum system has exactly one energetic representation. This is clearly assigned to the state. If the photon passes the beam splitter ( ST1 ), two components of the state must be assigned to the photon based on the principle of superposition. The energy assigned to the photon can now either on the way ( 2nd ) or on the way ( 3rd ) to the beam splitter ( ST2 ) reach. After the beam splitter ( ST1 ) two energetic representations are therefore possible in principle. On the one hand, the energy assigned to the photon can be located in the spatial area that is assigned to the reflected state component (path ( 3rd )). On the other hand, the energy assigned to the photon can be located in the spatial area that is assigned to the transmitted state component (path (2)). In principle, it is impossible to decide which option was actually implemented. Are the ways ( 2nd ) and ( 3rd ) of exactly the same length, the photon will certainly be on the detector ( D1 ) proven. Regardless of which energetic representation was realized.

Now bring Alice your absorber ( FROM ) in the beam path (blocks the path (2)), there are two cases: In the first case, if the energy assigned to the photon is located in the spatial area that is assigned to the transmitted state component, the energy of the photon (and thus the photon) is absorbed by the absorber. As in the classic case, this process halves the total number of photons detected by the two detectors per time. In the second case, if the energy assigned to the photon is located in the spatial area that is assigned to the reflected state component, the energy of the photon (and thus the photon) is detected by the detector in 50% of the cases ( D1 ) and in 50% of the cases by the detector ( D2 ) proven. Whenever the detector ( D2 ), it can now be said with certainty that the path ( 2nd ) is blocked. The prerequisite for this, however, is that an energy exchange between the space area of Alice and that of Bob is fundamentally possible and actually takes place in 50% of the cases when both beam paths are available. This is explained in detail in [ 11 ] proposed methods for interaction-free quantum measurement in [ 1 ] discussed.

Within the framework of classical physics, information can only be transmitted if energy is transmitted from the transmitter to the receiver. Regardless of how the information is transmitted. Whether you write the information on a piece of paper, wrap it around a stone and throw the stone at the recipient, or use light to transmit information, energy must always be transferred from the sender of the information to the recipient of the information. According to the special theory of relativity [ 18th ] the speed of light c has the same value in all inertial systems and matter can only be accelerated to a velocity v, with v <c, information can thus only be transmitted within the framework of classical physics at the maximum speed of light.

However, it cannot be concluded from this that information transmission is in principle only possible at the maximum speed of light [19]. This is easy to see. The special theory of relativity is based on two postulates: I. .) All natural laws are covariant with regard to the transformations between inertial systems. II .) The speed of light c has the same value in all inertial systems. From the principle of relativity (postulate I. ) it can only be concluded that there must be an invariant speed with regard to the permissible transformations between the inertial systems. It cannot be deduced from the principle of relativity that this invariant speed, in whatever sense, represents a fundamental upper limit for an information transfer. It is also not possible to derive the concrete value for this invariant speed from the relativity principle. First about postulate II this invariant speed becomes a concrete value - the speed of light c - assigned [ 20 ].

From the special theory of relativity, it can only be derived that information that is transmitted by means of an energy transfer can be transmitted at the maximum speed of light. However, it cannot be deduced from the special theory of relativity that information transfer in principle is only possible at the maximum speed of light.

Quantum physics differs fundamentally from classical physics. In classic physics, a cause can be given for each effect. However, this is no longer the case in quantum physics. As early as 1935, Einstein, Podolsky and Rosen (EPR) were the first to point out that quantum systems can interact with one another instantaneously over any distance, and that quantum physics can neither give a cause nor a plausible explanation for these effects [ 16 ]. This non-locality occurring in the context of quantum physics contradicts the locality concept of classical physics (Einstein locality) [ 21st ]. Quantum physics cannot therefore be traced back to a local theory in the sense of Einstein locality.

Since (EPR) the question has arisen as to whether an instantaneous transfer of information is possible within the framework of quantum physics? It is known that a single EPR source which, for example, simultaneously has two photons entangled in the degree of freedom of polarization (photon A and photon B ) can not be used to transmit a message. This is due to the no-signaling theorem valid in quantum physics. In its simplest form, this theorem states that no measurement on Photon, however carried out by the transmitter (Alice), is carried out A the statistical distribution of the possible measurement results to Photon B can be changed at the receiver (bob). It could also be shown that the statistical distribution of the possible measurement results in Bob can not be changed by any unitary state transformation, the effect of which is limited on the subsystem accessible to Alice.

However, it cannot be concluded from this that the no-signaling theorem makes it impossible in principle to transmit information using entangled quantum systems because the theorem does not cover all conceivable cases. It does not take into account the fact that there are physically realizable unitary state transformations, the effect of which is not limited to the subsystem accessible to Alice, although the physical interventions are carried out exclusively on the subsystem accessible to Alice.

• Symmetrisches Phasengitter: Unter einem Symmetrischen Phasengitter (SPG) wird eine Anordnung verstanden, welche wie ein Phasengitter, vorzugsweise ein verlustfreies Phasengitter, wirkt und bei dem parallel zur Gitternormalen einfallendes Licht der Wellenlänge A zu 50% in die N-te Beugungsordnung und 50% in die (- N)-te Beugungsordnung gebeugt wird (N = 1, 2, 3, ...).

The following definitions are used below:

• Symmetrical phase grating: A symmetrical phase grating (SPG) is understood to mean an arrangement which acts like a phase grating, preferably a loss-free phase grating, and with the light of the wavelength incident parallel to the grating normal A 50% is diffracted into the Nth diffraction order and 50% into the (- N) th diffraction order (N = 1, 2, 3, ...).

auf die Gitterkonstante G abgestimmt wird. Mit N = 1, 2, 3,.... und vorzugsweise a = β = 30 °. Wobei der eingeschlossene Winkel zwischen den Impulsmoden (-A0) und (-A1) 120° beträgt und die Impulsmoden parallel zur XZ-Ebene orientiert sind.In the simplest case, this can be a flat reflection grating with a (periodic) sawtooth-like profile with the period length G, which is designed such that both legs ( BLF0 ) and ( BLF1 ) of the sawtooth-like profile act as blaze surfaces and the blaze angle α is chosen to be the same size for both legs, preferably 30 °. The included angle between the two legs of the sawtooth-like profile (the two blaze surfaces) is then 120 ° (see here ). For example, if the flat reflection grating lies in the XY plane and points the grating normal in the Z direction and the blaze surfaces ( BLF0 ) and ( BLF1 ) oriented parallel to the Y axis, the sawtooth-like profile of the reflection grating is repeated with the period length G (Lattice constant) in the X direction. Parallel light with the wavelength A that strikes the reflection grating in the pulse mode (-AL) in the (-Z) direction is then bent 50% at an angle β of 30 ° to the X axis in the direction of the pulse mode (-A0) and 50 % at an angle β of 30 ° to the X axis in the direction of the pulse mode (-A1), provided the wavelength λ is in accordance with $$\text{cos}\left(\mathrm{\beta }\right)\text{G}=\text{N}\mathrm{\lambda }$$

to the lattice constant G is voted. With N = 1, 2, 3, .... and preferably a = β = 30 °. The included angle between the pulse modes (-A0) and (-A1) is 120 ° and the pulse modes are oriented parallel to the XZ plane.

According to the invention, the symmetrical phase grating (SPG) is operated inversely. That is, a photon ( P1 ) in pulse mode ( A1 ) and a photon ( P2 ) in pulse mode ( A0 ) meet the symmetrical phase grating ( SPG ) and are on this in the impulse mode ( AL ) bent. In the following it is assumed that the phase position of the pulse mode ( A0 ) and ( A1 ) incoming wave packets (photons) are matched to each other in the structure used so that both after diffraction at the symmetrical phase grating ( SPG ) in pulse mode ( AL ) are available.

Primary degree of freedom / Secondary degree of freedom: The pulse degree of freedom is referred to as the primary degree of freedom and all other degrees of freedom, such as the energetic degree of freedom or the degree of polarization are referred to as the secondary degrees of freedom.

First Type Measurement ( MT0 ): A measurement on the photons ( P1 ) and ( P2 ) is then used as a measurement of the first kind ( MT0 ) if this is carried out in such a way that it is in principle impossible to decide from which source the photons ( P1 ) and ( P2 ) were emitted.

Measurement of the second type ( MT1 ): A measurement on the photons ( P1 ) and ( P2 ) is then used as a measurement of the second type ( MT1 ) denotes if the information from which source the photons ( P1 ) and ( P2 ) were emitted after the measurement is retained.

According to the invention, it is proposed to use entangled photons for the interaction-free transfer of information (see here and ). For this, two identical sources ( Q1 ) and ( Q2 ) used, which simultaneously generate a pair of photons. A source ( Q1 ) two photons entangled in at least one secondary degree of freedom ( P1 ) and ( P3 ) generate and a source ( Q2 ) at the same time two photons entangled in at least one secondary degree of freedom ( P2 ) and (P4). The photons ( P1 ) to ( P4 ) are indicated by the indices 1 to 4th numbered consecutively. Secondary degrees of freedom are thus, for example, the degree of polarization or the degree of freedom in terms of energy. Two photons can be entangled in terms of their degree of energy freedom, for example, in that a source simultaneously generates two photons (one photon each in a pulse mode, the pulse modes should be distinguishable), which have a different wavelength. A photon with the wavelength λ _G and a photon with the wavelength λ _R . If it is not possible in principle to determine in which pulse mode the individual photons are emitted, the two photons are entangled in terms of their degree of energy freedom. The sender of the information (Alice) carries out a measurement of the first type (binary-coded information) ( MT0 ) using a symmetrical phase grating ( SPG ) at the photons ( P1 ) and ( P2 ) through which the photons ( P3 ) and ( P4 ) existing quantum system in a state of the first kind ( Z1 ) is transferred. This then encodes the first logical state, preferably the logical 0. Does Alice carry out a measurement of the second type ( MT1 ) at the photons ( P1 ) and ( P2 ), the photons ( P3 ) and ( P4 ) existing quantum system into a state of the second kind ( Z2 ) transferred and thereby the second logical state, preferably the logical one 1 , encoded. The states of the first kind ( Z1 ) and the second type ( Z2 ) can be both pure and mixed states. However, they must be distinguishable in terms of measurement technology. The measurement of the second type ( MT1 ) so that the information from which source the photons ( P1 ) and ( P2 ) after the measurement is retained and the measurement of the first type ( MT0 ) carried out in such a way that it is in principle impossible to decide from which source the photons ( P1 ) and ( P2 ) were emitted. The recipient of the information (Bob) can then determine the state of the photons ( P3 ) and ( P4 ) with regard to the property state of the first kind ( Z1 ) and condition of the second type ( Z2 ) using a symmetrical beam splitter (STB), analyze and recognize the currently transmitted logical state for the purpose of interaction-free information transmission. A message can then be transmitted bit by bit.

An advantageous embodiment of the invention provides that the degree of polarization freedom and / or the energetic degree of freedom is used as the secondary degree of freedom.

A further embodiment provides that for the state analysis of the photons ( P3 ) and ( P4 ) a beam splitter, preferably a symmetrical beam splitter (STB), is used and / or to implement the measurement of the first type ( MT0 ), at the photons ( P1 ) and ( P2 ), as a symmetrical phase grating ( SPG ) a flat, symmetrical reflection grating is used.

Advantageously, to implement the measurements ( MT0 ) and or ( MT1 ) polarizing beam splitters (if the photons are entangled in the degree of polarization freedom) and / or dichroic beam splitters (if the photons are entangled in the degree of energy freedom). A dichroic beam splitter refers to a beam splitter that is one of the two wavelengths λ _G , λ _R reflected and the other transmitted. For example, λ _G reflects and λ _R transmitted.

As related to and was shown for identical quantum systems, a strict distinction must be made between the terms "identical" and "indistinguishable". The statement two quantum systems are identical, refers to the properties assigned to the quantum systems, such as energy. The statement two quantum systems can be distinguished, refers to the targeted, repeatable manipulation of individual quantum systems. This is always ensured when clearly defined spatial areas can be assigned to the individual quantum systems so that they do not overlap. Two identical quantum systems are therefore indistinguishable if and only if the spatial areas assigned to them are identical (perfectly overlapping). For two identical, indistinguishable photons, the state assigned to the overall system must therefore be symmetrical (swapping the indices that characterize the two subsystems does not change the state).

In the example discussed above, if two identical, distinguishable photons are simultaneously on the symmetrical beam splitter ( ST ) meet, the incoming state Ψ ₁ at the beam splitter ( ST ) due to the classic beam splitter property (described by the unitary operator U _ST ) in the state Ψ ₂ transferred. The state Ψ ₂ cannot, however, be regarded as an expiring state, since the two photons after the beam splitter must be regarded as indistinguishable. According to the spin statistics theorem in this case, the expiring state (case III in the example above) must be symmetrical. Hence the condition Ψ ₂ are still symmetrized to maintain the state at the beam splitter ( ^B Ψ ₃ ).

The spin statistics theorem, however, does not only concern identical quantum systems. The spin statistics theorem relates generally to indistinguishable quantum systems of "the same kind". Whereby “the same variety” means the division into bosons and fermions. But not a subdivision with regard to the energy or the polarization state of the quantum systems involved.

The invention is explained in more detail below with the aid of an application example (see here ). Two identically structured sources ( Q1 ) and ( Q2 ) are to emit a pair of photons in the symmetrical Bell state Ψ ^{+, which} is entangled in the degree of polarization freedom, at freely selectable times. The photons should be linearly polarized. The two possible directions of polarization are designated with "H" for horizontal polarization (polarization lies in the beam plane) and with "V" for vertical polarization (polarization is perpendicular to the beam plane). Details on the implementation of corresponding sources can be found in [ 22 ]. The source ( Q1 ) the photons ( P1 ) and ( P3 ) and the source ( Q2 ) the photons ( P2 ) and ( P4 ) produce. All four photons are said to be identical in terms of energetic freedom. The photon ( P1 ) is said about the mirror ( SP1 ) in pulse mode " A1 “To the point ( PA ) on a lossless, symmetrical phase grating ( SPG ) and the photon ( P2 ) is said to be above the mirror ( SP2 ) in pulse mode " A0 “To the point ( PA ) on the symmetrical phase grating ( SPG ) reach. At the symmetrical phase grating ( SPG ) the photons ( P1 ) and ( P2 ) into fashion ( AL ) bent. A photon in pulse mode ( AL ) then hits the polarizing beam splitter (PTSPG). The polarizing beam splitter ( PTSPG ) is arranged so that a pulse mode ( AL ) incoming photon with horizontal polarization transmits it and from the detector ( DSPGH ) is detected and in the impulse mode ( AL ) incoming photon with vertical polarization is reflected on it and from the detector ( DSPGV ) is proven. The symmetrical phase grating ( SPG ) forms together with the polarizing beam splitter ( PTSPG ) and the detectors ( DSPGH ) and ( DSPGV ) the detector ( DSPG ) which is a measurement of the first kind ( MT0 ) enables.

$${\mathrm{\Psi }}_{\text{Q2}}=1/2^{1/2}|\text{B0}{>}_4|\text{A0}{>}_2\left(|\text{H}{>}_4|\text{V}{>}_2+|\text{V}{>}_4|\text{H}{>}_2\right)$$

zugeordnet werden. Für das aus den vier Photonen bestehende Gesamtsystem ergibt sich somit der Zustand $${\mathrm{\Psi }}_{\text{Q}1/2}=1/4^{1/2}|\text{B}0{>}_4|\text{B}1{>}_3|\text{A}0{>}_2|\text{A1}{>}_1\left(|\text{H}{>}_3|\text{V}{>}_{\text{1}}+|\text{V}{>}_3|\text{H}{>}_{\text{1}}\right)\left(|\text{H}{>}_4|\text{V}{>}_2+|\text{V}{>}_4|\text{H}{>}_2\right)$$

da die beiden Photonenpaare unabhängig voneinander erzeugt wurden.The photon ( P3 ) is said about the mirror ( SP3 ) in pulse mode " B1 “To the point ( PB ) on a symmetrical, lossless beam splitter ( STB ) and the photon ( P4 ) is said about the mirror ( SP4 ) in pulse mode " B0 “To the point ( PB ) on the symmetrical beam splitter ( STB ) reach. The photon pairs emitted by the sources can then after the four photons the mirrors ( SP1 ) to ( SP4 ) the states have happened $${\mathrm{\Psi }}_{\text{Q1}}=1/{\mathrm{<figure-callout\; id="1"\; label="2nd"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">2nd</figure-callout>}}^{1/\mathrm{2nd}}|\text{B1}{>}_{\mathrm{3rd}}|\text{A1}{>}_1\left(|\text{H}{>}_{\mathrm{3rd}}|\text{V}{>}_1+|\text{V}{>}_{\mathrm{3rd}}|\text{H}{>}_1\right)$$

$${\mathrm{\Psi }}_{\text{Q2}}=1/{\mathrm{<figure-callout\; id="1"\; label="2nd"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">2nd</figure-callout>}}^{1/\mathrm{2nd}}|\text{B0}{>}_{\mathrm{4th}}|\text{A0}{>}_{\mathrm{2nd}}\left(|\text{H}{>}_{\mathrm{4th}}|\text{V}{>}_{\mathrm{2nd}}+|\text{V}{>}_{\mathrm{4th}}|\text{H}{>}_{\mathrm{2nd}}\right)$$

be assigned. This results in the state of the overall system consisting of the four photons $${\mathrm{\Psi }}_{\text{Q}}$$$${1/\mathrm{2nd}}_{}=1/{\mathrm{<figure-callout\; id="1"\; label="4th"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">4th</figure-callout>}}^{1/\mathrm{2nd}}|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}|\text{<figure-callout id="1" label="B" filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png" state="{{state}}">B</figure-callout>}1{>}_{\mathrm{3rd}}|\text{A}0{>}_{\mathrm{2nd}}|\text{A1}{>}_1\left(|\text{H}{>}_{\mathrm{3rd}}|\text{V}{>}_{\text{1}}+|\text{V}{>}_{\mathrm{3rd}}|\text{H}{>}_{\text{1}}\right)\left(|\text{H}{>}_{\mathrm{4th}}|\text{V}{>}_{\mathrm{2nd}}+|\text{V}{>}_{\mathrm{4th}}|\text{H}{>}_{\mathrm{2nd}}\right)$$

because the two pairs of photons were generated independently.

If the symmetrical phase grating ( SPG ) from the beam path, a photon in the pulse mode ( A1 ) now the polarizing beam splitter ( PTA2 ) to reach. The polarizing beam splitter ( PTA2 ) is arranged so that a pulse mode ( A1 ) incoming photon with horizontal polarization transmits it and from the detector ( DA2H ) is detected and in the impulse mode ( A1 ) incoming photon with vertical polarization is reflected on it and from the detector ( DA4V ) is proven. A photon in pulse mode ( A0 ) with the symmetrical phase grating removed ( SPG ) now the polarizing beam splitter ( PTA1 ) to reach. The polarizing beam splitter ( PTA1 ) is arranged so that a pulse mode ( A0 ) incoming photon with horizontal polarization transmits it and from the detector ( DA1H ) is detected and in the impulse mode ( A0 ) incoming photon with vertical polarization is reflected on it and from the detector ( DA3V ) is proven. The distance of the source ( Q1 ) to the point ( PA ) and the distance of the source ( Q2 ) to the point ( PA ) should be the same size so that the photons ( P1 ) and ( P2 ) at the same time at the point ( PA ) arrive. The distance of the source ( Q1 ) to the point ( PB ) and the distance of the source ( Q2 ) to the point ( PB ) have the same size so that the photons ( P3 ) and ( P4 ) at the same time at the point ( PB ) arrive. The distance of the sources ( Q1 ) and ( Q2 ) to the point ( PB ) should be larger than the distance between the sources ( Q1 ) and ( Q2 ) to the detectors ( DA1H ), ( DA2H ), ( DA3V ), DA4V ) and ( DSPG ). Photons that follow the symmetrical beam splitter ( STB ) in pulse mode ( B1 ) can be located using the detector ( DB2 ) be detected. Photons that follow the symmetrical beam splitter in the pulse mode ( B0 ) can be located using the detector ( DB1 ) be detected.

Using your arrangement, Alice now has two options for taking a measurement. 1 .) The symmetrical phase grating ( SPG ) is in the beam path as described: measurement "MT0". 2nd .) Alice removed ( SPG ) from the beam path: measurement "MT1".

1. A): Die Detektoren (DA3V) und (DA4V) detektieren je ein Photon. Alice weiß in diesem Fall, dass das Photon (P3) und das Photon (P4) horizontal polarisiert ist.
2. B): Die Detektoren (DA1H) und (DA2H) detektieren je ein Photon. Alice weiß in diesem Fall, dass das Photon (P3) und das Photon (P4) vertikal polarisiert ist.
3. C): Die Detektoren (DA1H) und (DA4V) detektieren je ein Photon. Alice weiß in diesem Fall, dass das Photon (P4) vertikal polarisiert ist und das Photon (P3) horizontal polarisiert ist.
4. D): Die Detektoren (DA3V) und (DA2H) detektieren je ein Photon. Alice weiß in diesem Fall, dass das Photon (P4) horizontal polarisiert ist und das Photon (P3) vertikal polarisiert ist.

Guide Alice to the photons ( P1 ) and ( P2 ) the measurement MT1 through (symmetrical phase grating ( SPG ) is removed from the beam path), the photons ( P1 ) and ( P2 ) by means of the detectors ( DA1H ), ( DA2H ), ( DA3V ) and ( DA4V ) to be analyzed. Four results are then possible:

1. A): The detectors ( DA3V ) and ( DA4V ) detect one photon each. In this case Alice knows that the photon ( P3 ) and the photon ( P4 ) is horizontally polarized.
2. B): The detectors ( DA1H ) and ( DA2H ) detect one photon each. In this case Alice knows that the photon ( P3 ) and the photon ( P4 ) is vertically polarized.
3. C): The detectors ( DA1H ) and ( DA4V ) detect one photon each. In this case Alice knows that the photon ( P4 ) is vertically polarized and the photon ( P3 ) is horizontally polarized.
4. D): The detectors ( DA3V ) and ( DA2H ) detect one photon each. In this case Alice knows that the photon ( P4 ) is horizontally polarized and the photon ( P3 ) is vertically polarized.

beschrieben werden. Wobei mit $$|\text{A}><\text{A}|,|\text{B}><\text{B}|,|\text{C}><\text{C}|\text{ und }|\text{D}><\text{D}|$$

die den reinen Zuständen $$\begin{array}{l}|\text{A}>=|\text{B0}{>}_4|\text{B1}{>}_3|\text{H}{>}_4|\text{H}{>}_3\\ |\text{B}>=|\text{B0}{>}_4|\text{B1}{>}_3|\text{V}{>}_4|\text{V}{>}_3\\ |\text{C}>=|\text{B0}{>}_4|\text{B1}{>}_3|\text{V}{>}_4|\text{V}{>}_3\\ |\text{D}>=|\text{B0}{>}_4|\text{B1}{>}_3|\text{H}{>}_4|\text{H}{>}_3\end{array}$$

zugeordneten Dichteoperatoren bezeichnet werden. Da im Fall (A) und (B) die Photonen (P3) und (P4) den Strahlteiler (STB) gleichzeitig erreichen und es in diesen Fällen prinzipiell unmöglich ist, zu entscheiden, ob die Photonen (P3) und (P4) am Strahlteiler (STB) reflektiert oder transmittiert wurden und es im Fall (C) und (D) aufgrund der über den Polarisationsfreiheitsgrad vorliegenden Informationen prinzipiell möglich ist, zu entscheiden, welches Photon am Strahlteiler (STB) reflektiert und welches transmittiert wurde [22], erhält man nach kurzer Rechnung für die am Strahlteiler (STB) auslaufenden Photonen (P3) und (P4) dann den Zustand $$\begin{array}{ll}|\text{MT1}/\text{OUT}><\text{MT1}/\text{OUT}|=\hfill & 1/4(|\text{A}/\text{OUT><A/OUT}|+|\text{B}/\text{OUT}><|\text{B}/\text{OUT}|\hfill \\ \hfill & +|\text{C}/\text{OUT}><\text{C}/\text{OUT}|+|\text{D}/\text{OUT}><\text{D}/\text{OUT}|).\hfill \end{array}$$

Wobei mit $$\begin{array}{l}|\text{A}/\text{OUT}><\text{A}/\text{OUT}|,|\text{B}/\text{OUT}><\text{B}/\text{OUT}|,|\text{C}/\text{OUT}><\text{C}/\text{OUT}|,\\ und|\text{D}/\text{OUT}><\text{D}/\text{OUT}|\end{array}$$

die den reinen Zuständen $$\begin{array}{l}|\text{A}/\text{OUT}>=\text{i}/2^{1/2}\left(\left|\text{B}0{>}_4\right|\text{B}0{>}_3+\left|\text{B1}{>}_4\right|\text{B1}{>}_3\right)\left|\text{H}{>}_4\right|\text{H}{>}_3\\ |\text{B}/\text{OUT}>=\text{i}/2^{1/2}\left(\left|\text{B}0{>}_4\right|\text{B}0{>}_3+\left|\text{B1}{>}_4\right|\text{B1}{>}_3\right)\left|\text{V}{>}_4\right|\text{V}{>}_3\\ |\text{C}/\text{OUT}>=\text{i}/4^{1/2}\left(\text{i}\left|\text{B}0{>}_4\right|\text{B}0{>}_3+\text{i}\left|\text{B1}{>}_4\right|\text{B1}{>}_3+\left|\text{B}0{>}_4\right|\text{B1}{>}_3-\left|\text{B1}{>}_4\right|\text{B}0{>}_3\right)\left|\text{V}{>}_4\right|\text{H}{>}_3\\ |\text{D}/\text{OUT}>=\text{i}/4^{1/2}\left(\text{i}\left|\text{B}0{>}_4\right|\text{B}0{>}_3+\text{i}\left|\text{B1}{>}_4\right|\text{B1}{>}_3+\left|\text{B}0{>}_4\right|\text{B1}{>}_3-\left|\text{B1}{>}_4\right|\text{B}0{>}_3\right)\left|\text{H}{>}_4\right|\text{H}{>}_3\end{array}$$

zugeordneten Dichteoperatoren bezeichnet werden.As long as Alice does not share your results, the photons arriving at Bob ( P3 ) and ( P4 ) by means of the mixed state $$|\text{MT1}><\text{MT1}|=1/\mathrm{4th}\left(|\text{A}><\text{A}|+|\text{B}><\text{B}|+|\text{C.}><\text{C.}|+|\text{D}><\text{D}|\right)$$

to be discribed. Whereby with $$|\text{A}><\text{A}|,|\text{B}><\text{B}|,|\text{C.}><\text{C.}|\text{and}|\text{D}><\text{D}|$$

the pure states $$\begin{array}{l}|\text{A}>=|\text{B0}{>}_{\mathrm{4th}}|\text{B1}{>}_{\mathrm{3rd}}|\text{H}{>}_{\mathrm{4th}}|\text{H}{>}_{\mathrm{3rd}}\\ |\text{B}>=|\text{B0}{>}_{\mathrm{4th}}|\text{B1}{>}_{\mathrm{3rd}}|\text{V}{>}_{\mathrm{4th}}|\text{V}{>}_{\mathrm{3rd}}\\ |\text{C.}>=|\text{B0}{>}_{\mathrm{4th}}|\text{B1}{>}_{\mathrm{3rd}}|\text{V}{>}_{\mathrm{4th}}|\text{V}{>}_{\mathrm{3rd}}\\ |\text{D}>=|\text{B0}{>}_{\mathrm{4th}}|\text{B1}{>}_{\mathrm{3rd}}|\text{H}{>}_{\mathrm{4th}}|\text{H}{>}_{\mathrm{3rd}}\end{array}$$

assigned density operators. Since in the case ( A ) and ( B ) the photons ( P3 ) and ( P4 ) the beam splitter ( STB ) at the same time and in these cases it is in principle impossible to decide whether the photons ( P3 ) and ( P4 ) on the beam splitter ( STB ) reflected or transmitted and in the case ( C. ) and ( D ) Based on the information available on the degree of polarization freedom, it is possible in principle to decide which photon on the beam splitter ( STB ) reflected and which was transmitted [ 22 ], you get after a short calculation for those on the beam splitter ( STB ) expiring photons ( P3 ) and ( P4 ) then the state $$\begin{array}{ll}|\text{MT1}/\text{OUT}><\text{MT1}/\text{OUT}|=\hfill & 1/\mathrm{4th}(|\text{A}/\text{OUT><A / OUT}|+|\text{B}/\text{OUT}><|\text{B}/\text{OUT}|\hfill \\ \hfill & +|\text{C.}/\text{OUT}><\text{C.}/\text{OUT}|+|\text{D}/\text{OUT}><\text{D}/\text{OUT}|).\hfill \end{array}$$

Whereby with $$\begin{array}{l}|\text{A}/\text{OUT}><\text{A}/\text{OUT}|,|\text{B}/\text{OUT}><\text{B}/\text{OUT}|,|\text{C.}/\text{OUT}><\text{C.}/\text{OUT}|,\\ und|\text{D}/\text{OUT}><\text{D}/\text{OUT}|\end{array}$$

the pure states $$\begin{array}{l}|\text{A}/\text{OUT}>=\text{i}/{\mathrm{2nd}}^{1/\mathrm{2nd}}\left(\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}+\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\right)\left|\text{H}{>}_{\mathrm{4th}}\right|\text{H}{>}_{\mathrm{3rd}}\\ |\text{B}/\text{OUT}>=\text{i}/{\mathrm{2nd}}^{1/\mathrm{2nd}}\left(\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}+\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\right)\left|\text{V}{>}_{\mathrm{4th}}\right|\text{V}{>}_{\mathrm{3rd}}\\ |\text{C.}/\text{OUT}>=\text{i}/{\mathrm{4th}}^{1/\mathrm{2nd}}\left(\text{i}\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}+\text{i}\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}+\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}-\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}\right)\left|\text{V}{>}_{\mathrm{4th}}\right|\text{H}{>}_{\mathrm{3rd}}\\ |\text{D}/\text{OUT}>=\text{i}/{\mathrm{4th}}^{1/\mathrm{2nd}}\left(\text{i}\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}+\text{i}\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}+\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}-\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}\right)\left|\text{H}{>}_{\mathrm{4th}}\right|\text{H}{>}_{\mathrm{3rd}}\end{array}$$

assigned density operators.

Bob's using the detectors ( DB1 ) and ( DB2 ) detected photons are strictly correlated with respect to the pulse modes in 75% of the cases (both photons are either in the pulse mode ( B0 ) and are detected by the detector ( DB1 ) detected, or are in the impulse mode ( B1 ) and are detected by the detector ( DB2 ) detected). In 25% of the cases, coincidence events (the detectors ( DB1 ) and ( DB2 ) detect one photon at a time).

• Nachdem die Photonen (P1) und (P2) an dem Symmetrischen Phasengitter (SPG) gebeugt wurden, verlassen diese das Symmetrische Phasengitter (SPG) in der Impulsmode (AL). Hierdurch wird der Zustand Ψ_Q1/2 (Gl. 19) in den Zustand

$${\mathrm{\Psi }}_{\text{SPG}}=1/4^{1/2}|\text{B0}{>}_4|\text{B1}{>}_3|\text{AL}{>}_2|\text{AL}{>}_1\left(|\text{H}{>}_3|\text{V}{>}_1+|\text{V}{>}_3|\text{H}{>}_1\right)\left(|\text{H}{>}_4|\text{V}{>}_2+|\text{V}{>}_4|\text{H}{>}_2\right)$$

überführt.Leads Alice to the in schematically illustrated arrangement on the photons ( P1 ) and ( P2 ) the measurement MT0 through (symmetrical phase grating ( SPG ) is in the beam path as described), the following situation arises:

• After the photons ( P1 ) and ( P2 ) on the symmetrical phase grating ( SPG ) have been bent, they leave the symmetrical phase grating ( SPG ) in pulse mode ( AL ). This will make the condition Ψ _{Q1 / 2} (Eq. 19) in the state

$${\mathrm{\Psi }}_{\text{SPG}}=1/{\mathrm{<figure-callout\; id="1"\; label="4th"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">4th</figure-callout>}}^{1/\mathrm{2nd}}|\text{B0}{>}_{\mathrm{4th}}|\text{B1}{>}_{\mathrm{3rd}}|\text{AL}{>}_{\mathrm{2nd}}|\text{AL}{>}_1\left(|\text{H}{>}_{\mathrm{3rd}}|\text{V}{>}_1+|\text{V}{>}_{\mathrm{3rd}}|\text{H}{>}_1\right)\left(|\text{H}{>}_{\mathrm{4th}}|\text{V}{>}_{\mathrm{2nd}}+|\text{V}{>}_{\mathrm{4th}}|\text{H}{>}_{\mathrm{2nd}}\right)$$

transferred.

Since after the photons ( P1 ) and ( P2 ) on the symmetrical phase grating ( SPG ) have been diffracted, it is not even possible in principle to decide from which source the individual photons were emitted, the state Ψ _SPG (Eq. 26) regarding the indices 1 and 2nd be symmetrized. After the symmetrization you get the state $$\begin{array}{l}{\mathrm{\Psi }}_{\text{THERE}}=1/{\mathrm{<figure-callout\; id="1"\; label="4th"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">4th</figure-callout>}}^{1/\mathrm{2nd}}|\text{B0}{>}_{\mathrm{4th}}|\text{B1}{>}_{\mathrm{3rd}}|\text{AL}{>}_{\mathrm{2nd}}|\text{AL}{>}_1[1/{\mathrm{<figure-callout\; id="1"\; label="2nd"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">2nd</figure-callout>}}^{1/\mathrm{2nd}}\left(|\text{H}{>}_{\mathrm{2nd}}|\text{V}{>}_1+\text{V}{>}_{\mathrm{2nd}}|\text{H}{>}_1\right)\left(|\text{H}{>}_{\mathrm{4th}}\text{V}{>}_{\mathrm{3rd}}+\text{V}{>}_{\mathrm{4th}}|\text{H}{>}_{\mathrm{3rd}}\right)\\ +|\text{H}{>}_{\mathrm{4th}}|\text{H}{>}_{\mathrm{3rd}}|\text{V}{>}_{\mathrm{2nd}}|\text{V}{>}_1+|\text{V}{>}_{\mathrm{4th}}|\text{V}{>}_{\mathrm{3rd}}|\text{H}{>}_{\mathrm{2nd}}|\text{H}{>}_1]\end{array}$$

• (E): Auf den Detektor (DSPG) wird bei der Detektion der Photonen (P1) und (P2) die Information übertragen, dass beide Photonen vertikal polarisiert sind.
• (F): Auf den Detektor (DSPG) wird bei der Detektion der Photonen (P1) und (P2) die Information übertragen, dass beide Photonen horizontal polarisiert sind.
• (G): Auf den Detektor (DSPG) wird bei der Detektion der Photonen (P1) und (P2) die Information übertragen, dass ein Photonen horizontal polarisiert ist und ein Photon vertikal polarisiert ist. Wobei es nicht einmal im Prinzip möglich ist, zu entscheiden welches Photon horizontal und welches vertikal polarisiert ist.

This means that three events are possible:

• (E): On the detector ( DSPG ) is used in the detection of the photons ( P1 ) and ( P2 ) transmit the information that both photons are vertically polarized.
• (F): On the detector ( DSPG ) is used in the detection of the photons ( P1 ) and ( P2 ) transmit the information that both photons are horizontally polarized.
• ( G ): On the detector ( DSPG ) is used in the detection of the photons ( P1 ) and ( P2 ) transmit the information that a photon is horizontally polarized and a photon is vertically polarized. In principle, it is not even possible to decide which photon is horizontally and which is polarized vertically.

zugeordnet werden. Wobei mit $$\left|\text{E}><\text{E}\right|,\left|\text{F}><\text{F}\right|\text{ und }\left|\text{G}><\text{G}\right|$$

die den reinen Zuständen $$\begin{array}{l}|\text{E}>=|\text{B0}{>}_4|\text{B1}{>}_3|\text{H}{>}_4|\text{H}{>}_3\\ |\text{F}>=|\text{B0}{>}_4|\text{B1}{>}_3|\text{V}{>}_4|\text{V}{>}_3\\ |\text{G}>=1/2^{1/2}|\text{B0}{>}_4|\text{B1}{>}_3\left(|\text{H}{>}_4|\text{V}{>}_3+|\text{V}{>}_4|\text{H}{>}_3\right)\end{array}$$

zugeordneten Dichteoperatoren bezeichnet werden. Nachdem die Photonen (P3) und (P4) den Strahlteiler (STB) passiert haben, erhält man dann nach kurzer Rechnung für diese den gemischten Zustand $$\begin{array}{c}\left|\text{MT0/OUT><MT0/OUT}\right|=1/4\left|\text{E/OUT><E/OUT}\right|+1/4|\text{F/OUT>}\text{<F/OUT}|\\ +1/2\left|\text{G/OUT><G/OUT}\right|.\end{array}$$

Since a measurement must be formally assigned to the projection on the state component that corresponds to the knowledge that is generally accessible (during the measurement), the photons arriving at Bob ( P3 ) and ( P4 ) hence the mixed state $$\left|\text{MT0;}\Vert ><\text{MT0;}\Vert \right|=1/\mathrm{4th}\left|\text{E}><\text{E}\right|+1/\mathrm{4th}\left|\text{F}><\text{F}\right|+1/\mathrm{2nd}\left|\text{G}><\text{G}\right|$$

be assigned. Whereby with $$\left|\text{E}><\text{E}\right|,\left|\text{F}><\text{F}\right|\text{and}\left|\text{G}><\text{G}\right|$$

the pure states $$\begin{array}{l}|\text{E}>=|\text{B0}{>}_{\mathrm{4th}}|\text{B1}{>}_{\mathrm{3rd}}|\text{H}{>}_{\mathrm{4th}}|\text{H}{>}_{\mathrm{3rd}}\\ |\text{F}>=|\text{B0}{>}_{\mathrm{4th}}|\text{B1}{>}_{\mathrm{3rd}}|\text{V}{>}_{\mathrm{4th}}|\text{V}{>}_{\mathrm{3rd}}\\ |\text{G}>=1/{\mathrm{<figure-callout\; id="1"\; label="2nd"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">2nd</figure-callout>}}^{1/\mathrm{2nd}}|\text{B0}{>}_{\mathrm{4th}}|\text{B1}{>}_{\mathrm{3rd}}\left(|\text{H}{>}_{\mathrm{4th}}|\text{V}{>}_{\mathrm{3rd}}+|\text{V}{>}_{\mathrm{4th}}|\text{H}{>}_{\mathrm{3rd}}\right)\end{array}$$

assigned density operators. After the photons ( P3 ) and ( P4 ) the beam splitter ( STB ) have passed, you will then receive the mixed state for them after a short calculation $$\begin{array}{c}\left|\text{MT0 / OUT><MT0 / OUT}\right|=1/\mathrm{4th}\left|\text{E / OUT><E / OUT}\right|+1/\mathrm{4th}|\text{F / OUT>}\text{<F / OUT}|\\ +1/\mathrm{2nd}\left|\text{G / OUT><G / OUT}\right|.\end{array}$$

die den reinen Zuständen $$\begin{array}{l}|\text{E/OUT>}=\text{i}/2^{1/2}\left(|\text{B0}{>}_4|\text{B0}{>}_3+|\text{B1}{>}_4|\text{B1}{>}_3\right)|\text{H}{>}_4|\text{H}{>}_3\\ |\text{F/OUT>}=\text{i}/2^{1/2}\left(|\text{B0}{>}_4|\text{B0}{>}_3+|\text{B1}{>}_4|\text{B1}{>}_3\right)|\text{V}{>}_4|\text{V}{>}_3\\ |\text{G/OUT>}=\text{i}/4^{1/2}\left(|\text{B0}{>}_4|\text{B0}{>}_3+|\text{B1}{>}_4|\text{B1}{>}_3\right)\left(|\text{H}{>}_4|\text{V}{>}_3+|\text{V}{>}_4|\text{H}{>}_3\right)\end{array}$$

zugeordneten Dichteoperatoren bezeichnet werden.Whereby with $$\left|\text{E / OUT><E / OUT}\right|,\left|\text{F / OUT><F / OUT}\right|,\text{and}\left|\text{G / OUT><G / OUT}\right|$$

the pure states $$\begin{array}{l}|\text{E / OUT>}=\text{i}/{\mathrm{2nd}}^{1/\mathrm{2nd}}\left(|\text{B0}{>}_{\mathrm{4th}}|\text{B0}{>}_{\mathrm{3rd}}+|\text{B1}{>}_{\mathrm{4th}}|\text{B1}{>}_{\mathrm{3rd}}\right)|\text{H}{>}_{\mathrm{4th}}|\text{H}{>}_{\mathrm{3rd}}\\ |\text{F / OUT>}=\text{i}/{\mathrm{2nd}}^{1/\mathrm{2nd}}\left(|\text{B0}{>}_{\mathrm{4th}}|\text{B0}{>}_{\mathrm{3rd}}+|\text{B1}{>}_{\mathrm{4th}}|\text{B1}{>}_{\mathrm{3rd}}\right)|\text{V}{>}_{\mathrm{4th}}|\text{V}{>}_{\mathrm{3rd}}\\ |\text{G / OUT>}=\text{i}/{\mathrm{4th}}^{1/\mathrm{2nd}}\left(|\text{B0}{>}_{\mathrm{4th}}|\text{B0}{>}_{\mathrm{3rd}}+|\text{B1}{>}_{\mathrm{4th}}|\text{B1}{>}_{\mathrm{3rd}}\right)\left(|\text{H}{>}_{\mathrm{4th}}|\text{V}{>}_{\mathrm{3rd}}+|\text{V}{>}_{\mathrm{4th}}|\text{H}{>}_{\mathrm{3rd}}\right)\end{array}$$

assigned density operators.

Bob's using the detectors ( DB1 ) and ( DB2 ) detected photons are in this case strictly correlated with regard to the pulse modes (both photons are either in the pulse mode ( B0 ) and are detected by the detector ( DB1 ) detected, or are in the impulse mode ( B1 ) and are detected by the detector ( DB2 ) detected). Coincidence events (the detectors ( DB1 ) and ( DB2 ) have one photon at a time) do not occur.

Bob can use the frequency of coincidence events to see if Alice is taking the measurement MT0 or MT1 performs, although there is never an exchange of energy (an interaction) between Alice and Bob.

Therefore, only one protocol needs to be agreed for an interaction-free transmission of information. For transmission, it can be agreed, for example, that the measurement MT0 a logical "0" and the measurement MT1 corresponds to a logical "1". It is also agreed that exactly one µs transmission time is used per bit. The times at which Alice would like to transmit information are specified beforehand. For example, Alice can always decide again at full µs whether she wants to transmit a logical "0" or a logical "1". Emit the sources ( Q1 ) and ( Q2 ) for example 1000 pairs of photons per µs, Bob can safely decide within one µs whether Alice has a logical "0" or a "1" want to transfer. Alice could then transmit data at a data rate of up to 1 Mbit / s without interaction.

The symmetrical phase grating ( SPG ) ensures that it is not even possible in principle to decide from which source the photons were emitted after the symmetrical phase grating ( SPG ) in pulse mode ( AL ) have left.

The method presented here can of course also be implemented with circularly polarized photons. All you have to do in the polarizing beam splitters ( PTA1 ), ( PTA2 ) and ( PTSPG ) are replaced by suitable components. For example, through a beam splitter that transmits left-handed photons and reflects right-handed photons.

A preferred embodiment of the invention provides that the photons ( P1 ) to ( P4 ), if these are entangled in the degree of energy freedom, can be detected by means of detectors which make it possible to determine the energy of the photons (see here ). The in schematically shown structure corresponds essentially to that in shown structure. To the in schematically shown structure you get if you from the in construction shown the polarizing beam splitters ( PTA1 ) and ( PTA2 ) and the detectors ( DA3V ) and ( DA4V ) away. In the With ( DA1H ) or with ( DA2H ) designated detectors are in designated with (DA1) or with (DA2). The polarizing beam splitter ( PTSPG ) and the detector ( DSPGV ) away. The in with (DSPGH) detector is in With ( DAL ) designated. Otherwise it's in shown construction with the in shown identical. The sources ( Q1 ) and ( Q2 ) should generate two photons entangled in the degree of freedom of energy at the same time. One photon each with the wavelength λ _R and a photon with the wavelength λ _G , with λ _R ≠ λ _G. The energetic resolution of the detectors used should be so large that a photon with the wavelength λ _R clearly from a photon of wavelength λ _G can be distinguished and can be recognized whether two or only one photon are detected. Advantageously, the [ 23 ] described superconducting detectors can be used for this.

Analogous to (Eq. 19th ) then results for the quantum system consisting of the four photons after the photons pass the mirrors ( SP1 ) to ( SP4 ) have happened the state $$\begin{array}{c}{\mathrm{\Psi }}_{\mathrm{\lambda \; /}\text{<figure-callout id="1" label="Q" filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png" state="{{state}}">Q</figure-callout>}\mathrm{1/2}}=1/{\mathrm{<figure-callout\; id="1"\; label="4th"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">4th</figure-callout>}}^{1/\mathrm{2nd}}|\text{B0}{>}_{\mathrm{4th}}|\text{B1}{>}_{\mathrm{3rd}}|\text{A0}{>}_{\mathrm{2nd}}|\text{A1}{>}_1\\ \left(|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{2nd}}|{\mathrm{\lambda }}_{\text{R}}{>}_1+|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}|{\mathrm{\lambda }}_{\text{G}}{>}_1\right)\left(|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{2nd}}+|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{4th}}|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{2nd}}\right).\end{array}$$

Using your arrangement, Alice now has two options for taking a measurement. 1 .) Measurement of the first type: the symmetrical phase grating ( SPG ) is in the beam path as described: measurement " MT0 ". 2nd .) Measurement of the second type: Alice removed ( SPG ) from the beam path: measurement " MT1 ".

• Wenn die Photonen (P1) und (P2) an dem Symmetrischen Phasengitter (SPG) in die Impulsmode (AL) gebeugt werden, wird der Zustand Ψ_λ/Q1/2 (Gl. 34) in den Zustand

$$\begin{array}{c}{\mathrm{\Psi }}_{\mathrm{\lambda }/\text{LWL}}=1/4^{1/2}|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_4|\text{<figure-callout id="1" label="B" filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png" state="{{state}}">B</figure-callout>}1{>}_3\left|\text{AL}{>}_2\right|\text{AL}{>}_1\\ \left(\left|{\mathrm{\lambda }}_G{>}_3\right|{\mathrm{\lambda }}_{\text{R}}{>}_1+\left|{\mathrm{\lambda }}_{\text{R}}{>}_3\right|{\mathrm{\lambda }}_{\text{G}}{>}_1\right)\left(|{\mathrm{\lambda }}_{\text{G}}{>}_4|{\mathrm{\lambda }}_{\text{R}}{>}_2+|{\mathrm{\lambda }}_{\text{R}}{>}_4|{\mathrm{\lambda }}_{\text{G}}{>}_2\right).\end{array}$$

überführt. Da es, nachdem die Photonen (P1) und (P2) das Symmetrische Phasengitter (SPG) passiert haben, nicht einmal im Prinzip möglich ist, zu entscheiden, von welcher Quelle die einzelnen Photonen emittiert wurden, muss der Zustand Ψ_λ/LWL (Gl. 35) bezüglich der Indizes 1 und 2 symmetrisiert werden. Nach der Symmetrisierung erhält man dann den Zustand $$\begin{array}{c}{\mathrm{\Psi }}_{\mathrm{\lambda }/\text{DA}}=1/4^{1/2}\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_4\right|\text{<figure-callout id="1" label="B" filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png" state="{{state}}">B</figure-callout>}1{>}_3\left|\text{AL}{>}_2\right|\text{AL}{>}_1[1/2^{1/2}\left(|{\mathrm{\lambda }}_{\text{G}}{>}_2|{\mathrm{\lambda }}_{\text{R}}{>}_1+|{\mathrm{\lambda }}_{\text{R}}{>}_2|{\mathrm{\lambda }}_{\text{G}}{>}_1\right)\\ \left(|{\mathrm{\lambda }}_{\text{G}}{>}_4{|{\mathrm{\lambda }}_{\text{R}}>}_3+|{\mathrm{\lambda }}_{\text{R}}{>}_4|{\mathrm{\lambda }}_{\text{G}}{>}_3\right)+|{\mathrm{\lambda }}_{\text{G}}{>}_4|{\mathrm{\lambda }}_{\text{G}}{>}_3|{\mathrm{\lambda }}_{\text{R}}{>}_2|{\mathrm{\lambda }}_{\text{R}}{>}_1+|{\mathrm{\lambda }}_{\text{R}}{>}_4|{\mathrm{\lambda }}_{\text{R}}{>}_3|{\mathrm{\lambda }}_{\text{G}}{>}_2|{\mathrm{\lambda }}_{\text{G}}{>}_1].\end{array}$$

Guide Alice to the photons ( P1 ) and ( P2 ) the measurement MT0 through (the symmetrical phase grating ( SPG ) is in the beam path as described), the following situation arises:

• If the photons ( P1 ) and ( P2 ) on the symmetrical phase grating ( SPG ) in pulse mode ( AL ) will be bowed, the condition Ψ _{λ / Q1 / 2} (Eq. 34) in the state

$$\begin{array}{c}{\mathrm{\Psi }}_{\mathrm{\lambda }/\text{FO}}=1/{\mathrm{<figure-callout\; id="1"\; label="4th"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">4th</figure-callout>}}^{1/\mathrm{2nd}}|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}|\text{<figure-callout id="1" label="B" filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png" state="{{state}}">B</figure-callout>}1{>}_{\mathrm{3rd}}\left|\text{AL}{>}_{\mathrm{2nd}}\right|\text{AL}{>}_1\\ \left(\left|{\mathrm{\lambda }}_G{>}_{\mathrm{3rd}}\right|{\mathrm{\lambda }}_{\text{R}}{>}_1+\left|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_1\right)\left(|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{2nd}}+|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{4th}}|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{2nd}}\right).\end{array}$$

transferred. Since after the photons ( P1 ) and ( P2 ) the symmetrical phase grating ( SPG ) have happened, it is not even possible in principle to decide from which source the individual photons were emitted, the state Ψ _{λ / LWL} (Eq. 35) regarding the indices 1 and 2nd be symmetrized. After the symmetrization you get the state $$\begin{array}{c}{\mathrm{\Psi }}_{\mathrm{\lambda }/\text{THERE}}=1/{\mathrm{<figure-callout\; id="1"\; label="4th"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">4th</figure-callout>}}^{1/\mathrm{2nd}}\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{<figure-callout id="1" label="B" filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png" state="{{state}}">B</figure-callout>}1{>}_{\mathrm{3rd}}\left|\text{AL}{>}_{\mathrm{2nd}}\right|\text{AL}{>}_1[1/{\mathrm{<figure-callout\; id="1"\; label="2nd"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">2nd</figure-callout>}}^{1/\mathrm{2nd}}\left(|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{2nd}}|{\mathrm{\lambda }}_{\text{R}}{>}_1+|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{2nd}}|{\mathrm{\lambda }}_{\text{G}}{>}_1\right)\\ \left(|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}{|{\mathrm{\lambda }}_{\text{R}}>}_{\mathrm{3rd}}+|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{4th}}|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{3rd}}\right)+|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{3rd}}|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{2nd}}|{\mathrm{\lambda }}_{\text{R}}{>}_1+|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{4th}}|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{2nd}}|{\mathrm{\lambda }}_{\text{G}}{>}_1].\end{array}$$

$$\text{X}=\left[\begin{array}{l}{\text{X}}_1\hfill \\ {\text{X}}_2\hfill \\ {\text{X}}_3\hfill \\ {\text{X}}_4\hfill \\ {\text{X}}_5\hfill \\ {\text{X}}_6\hfill \end{array}\right]\begin{array}{l}|1>\hfill \\ |2>\hfill \\ |3>\hfill \\ |4>\hfill \\ |5>\hfill \\ |6>\hfill \end{array}$$

mit: $$\begin{array}{l}|1>=\left|\text{B}0{>}_4\right|\text{B1}{>}_3\left|\text{AL}{>}_2\right|\text{AL}{>}_1\left|{\mathrm{\lambda }}_{\text{G}}{>}_4\right|{\mathrm{\lambda }}_{\text{R}}{>}_3\left|{\mathrm{\lambda }}_{\text{R}}{>}_2\right|{\mathrm{\lambda }}_{\text{G}}{>}_1\\ |2>=\left|\text{B}0{>}_4\right|\text{B1}{>}_3\left|\text{AL}{>}_2\right|\text{AL}{>}_1\left|{\mathrm{\lambda }}_{\text{G}}{>}_4\right|{\mathrm{\lambda }}_{\text{R}}{>}_3\left|{\mathrm{\lambda }}_{\text{R}}{>}_2\right|{\mathrm{\lambda }}_{\text{G}}{>}_1\\ |3>=\left|\text{B}0{>}_4\right|\text{B1}{>}_3\left|\text{AL}{>}_2\right|\text{AL}{>}_1\left|{\mathrm{\lambda }}_{\text{G}}{>}_4\right|{\mathrm{\lambda }}_{\text{R}}{>}_3\left|{\mathrm{\lambda }}_{\text{R}}{>}_2\right|{\mathrm{\lambda }}_{\text{G}}{>}_1\\ |4>=\left|\text{B}0{>}_4\right|\text{B1}{>}_3\left|\text{AL}{>}_2\right|\text{AL}{>}_1\left|{\mathrm{\lambda }}_{\text{G}}{>}_4\right|{\mathrm{\lambda }}_{\text{R}}{>}_3\left|{\mathrm{\lambda }}_{\text{R}}{>}_2\right|{\mathrm{\lambda }}_{\text{G}}{>}_1\\ |5>=\left|\text{B}0{>}_4\right|\text{B1}{>}_3\left|\text{AL}{>}_2\right|\text{AL}{>}_1\left|{\mathrm{\lambda }}_{\text{G}}{>}_4\right|{\mathrm{\lambda }}_{\text{R}}{>}_3\left|{\mathrm{\lambda }}_{\text{R}}{>}_2\right|{\mathrm{\lambda }}_{\text{G}}{>}_1\\ |6>=\left|\text{B}0{>}_4\right|\text{B1}{>}_3\left|\text{AL}{>}_2\right|\text{AL}{>}_1\left|{\mathrm{\lambda }}_{\text{G}}{>}_4\right|{\mathrm{\lambda }}_{\text{R}}{>}_3\left|{\mathrm{\lambda }}_{\text{R}}{>}_2\right|{\mathrm{\lambda }}_{\text{G}}{>}_1\end{array}$$

The unitary transformation assigned to the symmetrization can be done using the unitary operator U _LWL describe. This can be defined on the subspace spanned by the state vectors | 1>, | 2>, | 3>, | 4>, | 5>, | 6> as follows: $${\text{U}}_{\text{FO}}=\frac{1}{\sqrt{\mathrm{2nd}}}\cdot \begin{array}{l}|1>\hfill \\ |\mathrm{2nd}>\hfill \\ |\mathrm{3rd}>\hfill \\ |\mathrm{4th}>\hfill \\ |5>\hfill \\ |6>\hfill \end{array}\stackrel{\begin{array}{llllll}<1|\hfill & <\mathrm{2nd}|\hfill & <\mathrm{3rd}|\hfill & <\mathrm{4th}|\hfill & <5|\hfill & <6|\hfill \end{array}}{\left[\begin{array}{llllll}0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill \\ 1\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill \\ 1\hfill & 0\hfill & 0\hfill & -1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & -1\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & \sqrt{\mathrm{<figure-callout\; id="0"\; label="2nd"\; filenames="DE102019008600A1\_0063.png"\; state="\{\{state\}\}">2nd</figure-callout>}}\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & \sqrt{\mathrm{2nd}}\hfill \end{array}\right]}$$

$$\text{X}=\left[\begin{array}{l}{\text{X}}_1\hfill \\ {\text{X}}_{\mathrm{2nd}}\hfill \\ {\text{X}}_{\mathrm{3rd}}\hfill \\ {\text{X}}_{\mathrm{4th}}\hfill \\ {\text{X}}_5\hfill \\ {\text{X}}_6\hfill \end{array}\right]\begin{array}{l}|1>\hfill \\ |\mathrm{2nd}>\hfill \\ |\mathrm{3rd}>\hfill \\ |\mathrm{4th}>\hfill \\ |5>\hfill \\ |6>\hfill \end{array}$$

With: $$\begin{array}{l}|1>=\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\left|\text{AL}{>}_{\mathrm{2nd}}\right|\text{AL}{>}_1\left|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}\left|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{2nd}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_1\\ |\mathrm{2nd}>=\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\left|\text{AL}{>}_{\mathrm{2nd}}\right|\text{AL}{>}_1\left|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}\left|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{2nd}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_1\\ |\mathrm{3rd}>=\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\left|\text{AL}{>}_{\mathrm{2nd}}\right|\text{AL}{>}_1\left|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}\left|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{2nd}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_1\\ |\mathrm{4th}>=\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\left|\text{AL}{>}_{\mathrm{2nd}}\right|\text{AL}{>}_1\left|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}\left|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{2nd}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_1\\ |5>=\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\left|\text{AL}{>}_{\mathrm{2nd}}\right|\text{AL}{>}_1\left|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}\left|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{2nd}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_1\\ |6>=\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\left|\text{AL}{>}_{\mathrm{2nd}}\right|\text{AL}{>}_1\left|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}\left|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{2nd}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_1\end{array}$$

Therefore: $${\mathrm{\Psi }}_{\mathrm{\lambda }/\text{THERE}}={\text{U}}_{\text{FO}}{\mathrm{\Psi }}_{\mathrm{\lambda }/\text{FO}}$$

• (E;λ): Auf den Detektor (DAL) wird bei der Detektion der Photonen (P1) und (P2) die Information übertragen, dass beide Photonen die Wellenlänge λ_R haben.
• (F;A): Auf den Detektor (DAL) wird bei der Detektion der Photonen (P1) und (P2) die Information übertragen, dass beide Photonen die Wellenlänge λ_G haben.
• (G;A): Auf den Detektor (DAL) wird bei der Detektion der Photonen (P1) und (P2) die Information übertragen, dass ein Photonen die Wellenlänge λ_R und ein Photon die Wellenlänge λ_G hat. Wobei es nicht einmal im Prinzip möglich ist, zu entscheiden welchem Photon die Wellenlänge λ_R und welchem Photon die Wellenlänge λ_G zugeordnet werden muss.

Three events are now possible:

• (E; λ): On the detector ( DAL ) is used in the detection of the photons ( P1 ) and ( P2 ) transmit the information that both photons have the wavelength λ _R to have.
• (F; A): On the detector ( DAL ) is used in the detection of the photons ( P1 ) and ( P2 ) transmit the information that both photons have the wavelength λ _G to have.
• (G; A): On the detector ( DAL ) is used in the detection of the photons ( P1 ) and ( P2 ) transmit the information that a photon has the wavelength λ _R and a photon the wavelength λ _G Has. It is not even possible in principle to decide which photon the wavelength λ _R and which photon the wavelength λ _G must be assigned.

zugeordnet werden. Wobei mit $$\left|\text{E};\mathrm{\lambda }><\text{E};\mathrm{\lambda }\right|,\left|\text{F};\mathrm{\lambda }><\text{F};\mathrm{\lambda }\right|\text{und}\left|\text{G};\mathrm{\lambda ><}\text{G};\mathrm{\lambda }\right|$$

die den reinen Zuständen $$\begin{array}{l}|\text{E};\mathrm{\lambda >}=\left|\text{B}0{>}_4\right|\text{B1}{>}_3\left|{\mathrm{\lambda }}_{\text{G}}{>}_4\right|{\mathrm{\lambda }}_{\text{G}}{>}_3\\ |\text{F};\mathrm{\lambda >}=\left|\text{B}0{>}_4\right|\text{B1}{>}_3\left|{\mathrm{\lambda }}_{\text{R}}{>}_4\right|{\mathrm{\lambda }}_{\text{R}}{>}_3\\ |\text{G};\mathrm{\lambda >}=1/2^{1/2}\left|\text{B}0{>}_4\right|\text{B1}{>}_3\left(\left|{\mathrm{\lambda }}_{\text{G}}{>}_4\right|{\mathrm{\lambda }}_{\text{R}}{>}_3+\left|{\mathrm{\lambda }}_{\text{R}}{>}_4\right|{\mathrm{\lambda }}_{\text{G}}{>}_3\right)\end{array}$$

zugeordneten Dichteoperatoren bezeichnet werden. Nachdem die Photonen (P3) und (P4) den Strahlteiler (STB) passiert haben, erhält man dann nach kurzer Rechnung für diese den gemischten Zustand $$\begin{array}{l}\left|\text{MT}0/\text{OUT};\mathrm{\lambda ><}\text{MT}0/\text{OUT};\mathrm{\lambda }\right|=1/4\left|\text{E}/\text{OUT};\mathrm{\lambda ><}\text{E}/\text{OUT};\mathrm{\lambda }\right|\\ +1/4\left|\text{F}/\text{OUT};\mathrm{\lambda ><}\text{F}/\text{OUT};\mathrm{\lambda }\right|+1/2\left|\text{G}/\text{OUT};\mathrm{\lambda ><}\text{G}/\text{OUT};\mathrm{\lambda }\right|.\end{array}$$

Since a measurement must be formally assigned to the projection on the state component that corresponds to the knowledge that is generally accessible (during the measurement), the photons arriving at Bob ( P3 ) and ( P4 ) thus the mixed state of the first kind ( Z1 ) $$\left|\text{<figure-callout id="0" label="MT" filenames="DE102019008600A1\_0063.png" state="{{state}}">MT</figure-callout>}0;\mathrm{\lambda ><}\text{<figure-callout id="0" label="MT" filenames="DE102019008600A1\_0063.png" state="{{state}}">MT</figure-callout>}0;\mathrm{\lambda }\right|=1/\mathrm{4th}\left|\text{E};\mathrm{\lambda }><\text{E};\mathrm{\lambda }\right|+1/\mathrm{4th}\left|\text{F};\mathrm{\lambda ><}\text{F};\mathrm{\lambda }\right|+1/\mathrm{2nd}\left|\text{G};\mathrm{\lambda ><}\text{G};\mathrm{\lambda }\right|$$

be assigned. Whereby with $$\left|\text{E};\mathrm{\lambda }><\text{E};\mathrm{\lambda }\right|,\left|\text{F};\mathrm{\lambda }><\text{F};\mathrm{\lambda }\right|\text{and}\left|\text{G};\mathrm{\lambda ><}\text{G};\mathrm{\lambda }\right|$$

the pure states $$\begin{array}{l}|\text{E};\mathrm{\lambda >}=\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\left|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{3rd}}\\ |\text{F};\mathrm{\lambda >}=\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\left|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}\\ |\text{G};\mathrm{\lambda >}=1/{\mathrm{<figure-callout\; id="1"\; label="2nd"\; filenames="DE102019008600A1\_0063.png,DE102019008600A1\_0064.png"\; state="\{\{state\}\}">2nd</figure-callout>}}^{1/\mathrm{2nd}}\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\left(\left|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}+\left|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{3rd}}\right)\end{array}$$

assigned density operators. After the photons ( P3 ) and ( P4 ) the beam splitter ( STB ) have passed, you will then receive the mixed state for them after a short calculation $$\begin{array}{l}\left|\text{<figure-callout id="0" label="MT" filenames="DE102019008600A1\_0063.png" state="{{state}}">MT</figure-callout>}0/\text{OUT};\mathrm{\lambda ><}\text{<figure-callout id="0" label="MT" filenames="DE102019008600A1\_0063.png" state="{{state}}">MT</figure-callout>}0/\text{OUT};\mathrm{\lambda }\right|=1/\mathrm{4th}\left|\text{E}/\text{OUT};\mathrm{\lambda ><}\text{E}/\text{OUT};\mathrm{\lambda }\right|\\ +1/\mathrm{4th}\left|\text{F}/\text{OUT};\mathrm{\lambda ><}\text{F}/\text{OUT};\mathrm{\lambda }\right|+1/\mathrm{2nd}\left|\text{G}/\text{OUT};\mathrm{\lambda ><}\text{G}/\text{OUT};\mathrm{\lambda }\right|.\end{array}$$

die den reinen Zuständen $$\begin{array}{l}|\text{E}/\text{OUT};\mathrm{\lambda }>=\text{i}/2^{1/2}\left(\left|\text{B}0{>}_4\right|\text{B}0{>}_3+\left|\text{B1}{>}_4\right|\text{B1}{>}_3\right)\left|{\mathrm{\lambda }}_{\text{G}}{>}_4\right|{\mathrm{\lambda }}_{\text{G}}{>}_3\\ |\text{F}/\text{OUT};\mathrm{\lambda }>=\text{i}/2^{1/2}\left(\left|\text{B}0{>}_4\right|\text{B}0{>}_3+\left|\text{B1}{>}_4\right|\text{B1}{>}_3\right)\left|{\mathrm{\lambda }}_{\text{R}}{>}_4\right|{\mathrm{\lambda }}_{\text{R}}{>}_3\\ |\text{G}/\text{OUT};\mathrm{\lambda }>=\text{i}/4^{1/2}\left(\left|\text{B}0{>}_4\right|\text{B}0{>}_3+\left|\text{B1}{>}_4\right|\text{B1}{>}_3\right)(\left|{\mathrm{\lambda }}_{\text{G}}{>}_4\right|{\mathrm{\lambda }}_{\text{R}}{>}_3+\left|{\mathrm{\lambda }}_{\text{R}}{>}_5\right|{\mathrm{\lambda }}_{\text{G}}{>}_3)\end{array}$$

zugeordneten Dichteoperatoren bezeichnet werden.Whereby with $$\begin{array}{l}\left|\text{E}/\text{OUT};\mathrm{\lambda ><}\text{E}/\text{OUT};\mathrm{\lambda }\right|,\left|\text{F}/\text{OUT};\mathrm{\lambda ><}\text{F}/\text{OUT};\mathrm{\lambda }\right|\\ \text{and}\left|\text{G}/\text{OUT};\mathrm{\lambda ><}\text{G}/\text{OUT};\mathrm{\lambda }\right|\end{array}$$

the pure states $$\begin{array}{l}|\text{E}/\text{OUT};\mathrm{\lambda }>=\text{i}/{\mathrm{2nd}}^{1/\mathrm{2nd}}\left(\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}+\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\right)\left|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{3rd}}\\ |\text{F}/\text{OUT};\mathrm{\lambda }>=\text{i}/{\mathrm{2nd}}^{1/\mathrm{2nd}}\left(\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}+\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\right)\left|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}\\ |\text{G}/\text{OUT};\mathrm{\lambda }>=\text{i}/{\mathrm{4th}}^{1/\mathrm{2nd}}\left(\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}+\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\right)(\left|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}+\left|{\mathrm{\lambda }}_{\text{R}}{>}_5\right|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{3rd}})\end{array}$$

assigned density operators.

Bob's using the detectors ( DB1 ) and ( DB2 ) detected photons are thus strictly correlated with regard to the pulse modes (both photons are either in the pulse mode ( B0 ) and are detected by the detector ( DB1 ) detected, or are in the impulse mode ( B1 ) and are detected by the detector ( DB2 ) detected). Coincidence events (the detectors ( DB1 ) and ( DB2 ) have one photon at a time) do not occur.

• (A;λ): Die Detektoren (DA1) und (DA2) detektieren je ein Photon mit der Wellenlänge λ_R . Alice weiß in diesem Fall, dass dem Photon (P3) und dem Photon (P4) die Wellenlänge λ_G zugeordnet werden muss.
• (B;λ): Die Detektoren (DA1) und (DA2) detektieren je ein Photon mit der Wellenlänge λ_G . Alice weiß in diesem Fall, dass dem Photon (P3) und dem Photon (P4) die Wellenlänge λ_R zugeordnet werden muss.
• (C;A): Die Detektoren (DA1) und (DA2) detektieren je ein Photon. (DA1) detektiert ein Photon mit der Wellenlänge λ_R und (DA2) detektiert ein Photon mit der Wellenlänge λ_G . Alice weiß in diesem Fall, dass dem Photon (P4) die Wellenlänge λ_G und dem Photon (P3) die Wellenlänge λ_R zugeordnet werden muss.
• (D;A): Die Detektoren (DA1) und (DA2) detektieren je ein Photon. (DA1) detektiert ein Photon mit der Wellenlänge λ_G und (DA2) detektiert ein Photon mit der Wellenlänge λ_R . Alice weiß in diesem Fall, dass dem Photon (P4) die Wellenlänge λ_R und dem Photon (P3) die Wellenlänge λ_G zugeordnet werden muss.

Guide Alice to the photons ( P1 ) and ( P2 ) the measurement MT1 through (symmetrical phase grating ( SPG ) is removed from the beam path), the photons ( P1 ) and ( P2 ) can be analyzed using the detectors (DA1) and (DA2). Four results are then possible:

• ( A ; λ): The detectors ( DA1 ) and ( DA2 ) each detect a photon with the wavelength λ _R . In this case Alice knows that the photon ( P3 ) and the photon ( P4 ) the wavelength λ _G must be assigned.
• ( B ; λ): The detectors ( DA1 ) and ( DA2 ) each detect a photon with the wavelength λ _G . In this case Alice knows that the photon ( P3 ) and the photon ( P4 ) the wavelength λ _R must be assigned.
• ( C. ; A): The detectors ( DA1 ) and ( DA2 ) detect one photon each. ( DA1 ) detects a photon with the wavelength λ _R and ( DA2 ) detects a photon with the wavelength λ _G . In this case Alice knows that the photon ( P4 ) the wavelength λ _G and the photon ( P3 ) the wavelength λ _R must be assigned.
• ( D ; A): The detectors ( DA1 ) and ( DA2 ) detect one photon each. ( DA1 ) detects a photon with the wavelength λ _G and ( DA2 ) detects a photon with the wavelength λ _R . In this case Alice knows that the photon ( P4 ) the wavelength λ _R and the photon ( P3 ) the wavelength λ _G must be assigned.

beschrieben werden. Wobei mit $$\left|\text{A};\mathrm{\lambda ><}\text{A};\mathrm{\lambda }\right|,\left|\text{B};\mathrm{\lambda ><}\text{B};\mathrm{\lambda }\right|,\left|\text{C};\mathrm{\lambda ><}\text{C};\mathrm{\lambda }\right|\text{ und }\left|\text{D};\mathrm{\lambda ><}\text{D};\mathrm{\lambda }\right|$$

die den reinen Zuständen $$\begin{array}{l}|\text{A};\mathrm{\lambda >}=\left|\text{B}0{>}_4\right|\text{B1}{>}_3\left|{\mathrm{\lambda }}_{\text{G}}{>}_4\right|{\mathrm{\lambda }}_{\text{G}}{>}_3\\ |\text{B};\mathrm{\lambda >}=\left|\text{B}0{>}_4\right|\text{B1}{>}_3\left|{\mathrm{\lambda }}_{\text{R}}{>}_4\right|{\mathrm{\lambda }}_{\text{R}}{>}_3\\ |\text{C};\mathrm{\lambda >}=\left|\text{B}0{>}_4\right|\text{B1}{>}_3\left|{\mathrm{\lambda }}_{\text{G}}{>}_4\right|{\mathrm{\lambda }}_{\text{R}}{>}_3\\ |\text{D};\mathrm{\lambda >}=\left|\text{B}0{>}_4\right|\text{B1}{>}_3\left|{\mathrm{\lambda }}_{\text{R}}{>}_4\right|{\mathrm{\lambda }}_{\text{G}}{>}_3\end{array}$$

zugeordneten Dichteoperatoren bezeichnet werden. Da im Fall (A) und (B) die Photonen (P3) und (P4) den Strahlteiler (STB) gleichzeitig erreichen und es in diesen Fällen prinzipiell unmöglich ist, zu entscheiden, ob die Photonen (P3) und (P4) am Strahlteiler (STB) reflektiert oder transmittiert wurden und es im Fall (C) und (D) aufgrund der über den energetischen Freiheitsgrad vorliegenden Informationen prinzipiell möglich ist, zu entscheiden, welches Photon am Strahlteiler (STB) reflektiert und welches transmittiert wurde, erhält man nach kurzer Rechnung für die am Strahlteiler (STB) auslaufenden Photonen (P3) und (P4) dann den Zustand $$\begin{array}{l}\left|\text{MT1/OUT};\mathrm{\lambda ><}\text{MT1/OUT};\mathrm{\lambda }\right|=1/4(\left|\text{A}/\text{OUT};\mathrm{\lambda }><\text{A}/\text{OUT};\mathrm{\lambda }\right|+\left|\text{B}/\text{OUT};\mathrm{\lambda ><}\text{B}/\text{OUT};\mathrm{\lambda }\right|\\ +\left|\text{C}/\text{OUT};\mathrm{\lambda ><}\text{C}/\text{OUT};\mathrm{\lambda }\right|+\left|\text{D}/\text{OUT};\mathrm{\lambda ><}\text{D}/\text{OUT};\mathrm{\lambda }\right|).\end{array}$$

As long as Alice does not share your results, the photons arriving at Bob ( P3 ) and ( P4 ) by means of the mixed state of the second kind ( Z2 ) $$\begin{array}{c}\left|\text{MT1};\mathrm{\lambda ><}\text{MT1};\mathrm{\lambda }\right|=1/\mathrm{4th}(\left|\text{A};\mathrm{\lambda ><}\text{A};\mathrm{\lambda }\right|+\left|\text{B};\mathrm{\lambda ><}\text{B};\mathrm{\lambda }\right|\\ +\left|\text{C.};\mathrm{\lambda ><}\text{C.};\mathrm{\lambda }\right|+\left|\text{D};\mathrm{\lambda ><}\text{D};\mathrm{\lambda }\right|)\end{array}$$

to be discribed. Whereby with $$\left|\text{A};\mathrm{\lambda ><}\text{A};\mathrm{\lambda }\right|,\left|\text{B};\mathrm{\lambda ><}\text{B};\mathrm{\lambda }\right|,\left|\text{C.};\mathrm{\lambda ><}\text{C.};\mathrm{\lambda }\right|\text{and}\left|\text{D};\mathrm{\lambda ><}\text{D};\mathrm{\lambda }\right|$$

the pure states $$\begin{array}{l}|\text{A};\mathrm{\lambda >}=\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\left|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{3rd}}\\ |\text{B};\mathrm{\lambda >}=\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\left|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}\\ |\text{C.};\mathrm{\lambda >}=\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\left|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}\\ |\text{D};\mathrm{\lambda >}=\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\left|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{3rd}}\end{array}$$

assigned density operators. Since in the case ( A ) and ( B ) the photons ( P3 ) and ( P4 ) the beam splitter ( STB ) at the same time and in these cases it is in principle impossible to decide whether the photons ( P3 ) and ( P4 ) on the beam splitter ( STB ) reflected or transmitted and in the case ( C. ) and ( D ) Based on the information available on the degree of energy freedom, it is in principle possible to decide which photon on the beam splitter ( STB ) reflected and which one was transmitted can be obtained after a short calculation for those on the beam splitter ( STB ) expiring photons ( P3 ) and ( P4 ) then the state $$\begin{array}{l}\left|\text{MT1 / OUT};\mathrm{\lambda ><}\text{MT1 / OUT};\mathrm{\lambda }\right|=1/\mathrm{4th}(\left|\text{A}/\text{OUT};\mathrm{\lambda }><\text{A}/\text{OUT};\mathrm{\lambda }\right|+\left|\text{B}/\text{OUT};\mathrm{\lambda ><}\text{B}/\text{OUT};\mathrm{\lambda }\right|\\ +\left|\text{C.}/\text{OUT};\mathrm{\lambda ><}\text{C.}/\text{OUT};\mathrm{\lambda }\right|+\left|\text{D}/\text{OUT};\mathrm{\lambda ><}\text{D}/\text{OUT};\mathrm{\lambda }\right|).\end{array}$$

die den reinen Zuständen $$\begin{array}{l}|\text{A}/\text{OUT};\mathrm{\lambda }>=\text{i}/2^{1/2}\left(\left|\text{B}0{>}_4\right|\text{B}0{>}_3+\left|\text{B1}{>}_4\right|\text{B1}{>}_3\right)\left|{\mathrm{\lambda }}_{\text{G}}{>}_4\right|{\mathrm{\lambda }}_{\text{G}}{>}_3\\ |\text{B}/\text{OUT};\mathrm{\lambda }>=\text{i}/2^{1/2}\left(\left|\text{B}0{>}_4\right|\text{B}0{>}_3+\left|\text{B1}{>}_4\right|\text{B1}{>}_3\right)\left|{\mathrm{\lambda }}_{\text{R}}{>}_4\right|{\mathrm{\lambda }}_{\text{R}}{>}_3\\ |\text{C}/\text{OUT};\mathrm{\lambda }>=\text{i}/4^{1/2}\left(\text{i}\left|\text{B}0{>}_4\right|\text{B}0{>}_3+\text{i}\left|\text{B1}{>}_4\right|\text{B1}{>}_3+\left|\text{B}0{>}_4\right|\text{B1}{>}_3-\left|\text{B1}{>}_4\right|\text{B}0{>}_3\right)\left|{\mathrm{\lambda }}_{\text{G}}{>}_4\right|{\mathrm{\lambda }}_{\text{G}}{>}_3\\ |\text{D}/\text{OUT};\mathrm{\lambda }>=\text{i}/4^{1/2}\left(\text{i}\left|\text{B}0{>}_4\right|\text{B}0{>}_3+\text{i}\left|\text{B1}{>}_4\right|\text{B1}{>}_3+\left|\text{B}0{>}_4\right|\text{B1}{>}_3-\left|\text{B1}{>}_4\right|\text{B}0{>}_3\right)\left|{\mathrm{\lambda }}_{\text{R}}{>}_4\right|{\mathrm{\lambda }}_{\text{G}}{>}_3\end{array}$$

zugeordneten Dichteoperatoren bezeichnet werden. Die von Bob mittels der Detektoren (DB1) und (DB2) detektierten Photonen sind nun hinsichtlich der Impulsmoden nur noch in 75% der Fälle streng korreliert (beide Photonen befinden sich entweder in der Impulsmode (B0) und werden von dem Detektor (DB1) detektiert, oder befinden sich in der Impulsmode (B1) und werden von dem Detektor (DB2) detektiert). In 25% der Fälle werden nun Koinzidenzereignisse (die Detektoren (DB1) und (DB2) weisen gleichzeitig je ein Photon nach) auftreten. Bob kann somit anhand der Häufigkeit von Koinzidenzereignissen erkennen, ob Alice die Messung MT0 oder MT1 durchführt, obwohl es zu keinem Zeitpunkt einen Austausch von Energie (eine Wechselwirkung) zwischen Alice und Bob gibt.Whereby with $$\begin{array}{l}\left|\text{A}/\text{OUT};\mathrm{\lambda ><}\text{A}/\text{OUT};\mathrm{\lambda }\right|,\left|\text{B}/\text{OUT};\mathrm{\lambda ><}\text{B}/\text{OUT};\mathrm{\lambda }\right|,\left|\text{C.}/\text{OUT};\mathrm{\lambda ><}\text{C.}/\text{OUT};\mathrm{\lambda }\right|\\ \text{and}\left|\text{D}/\text{OUT};\mathrm{\lambda ><}\text{D}/\text{OUT};\mathrm{\lambda }\right|\end{array}$$

the pure states $$\begin{array}{l}|\text{A}/\text{OUT};\mathrm{\lambda }>=\text{i}/{\mathrm{2nd}}^{1/\mathrm{2nd}}\left(\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}+\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\right)\left|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{3rd}}\\ |\text{B}/\text{OUT};\mathrm{\lambda }>=\text{i}/{\mathrm{2nd}}^{1/\mathrm{2nd}}\left(\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}+\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}\right)\left|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{3rd}}\\ |\text{C.}/\text{OUT};\mathrm{\lambda }>=\text{i}/{\mathrm{4th}}^{1/\mathrm{2nd}}\left(\text{i}\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}+\text{i}\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}+\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}-\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}\right)\left|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{3rd}}\\ |\text{D}/\text{OUT};\mathrm{\lambda }>=\text{i}/{\mathrm{4th}}^{1/\mathrm{2nd}}\left(\text{i}\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}+\text{i}\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}+\left|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{4th}}\right|\text{B1}{>}_{\mathrm{3rd}}-\left|\text{B1}{>}_{\mathrm{4th}}\right|\text{<figure-callout id="0" label="B" filenames="DE102019008600A1\_0063.png" state="{{state}}">B</figure-callout>}0{>}_{\mathrm{3rd}}\right)\left|{\mathrm{\lambda }}_{\text{R}}{>}_{\mathrm{4th}}\right|{\mathrm{\lambda }}_{\text{G}}{>}_{\mathrm{3rd}}\end{array}$$

assigned density operators. Bob's using the detectors ( DB1 ) and ( DB2 ) detected photons are now strictly correlated with respect to the pulse modes only in 75% of the cases (both photons are either in the pulse mode ( B0 ) and are detected by the detector ( DB1 ) detected, or are in the impulse mode ( B1 ) and are detected by the detector ( DB2 ) detected). In 25% of the cases, coincidence events (the detectors ( DB1 ) and ( DB2 ) detect one photon at a time). Bob can use the frequency of coincidence events to see if Alice is taking the measurement MT0 or MT1 performs, although there is never an exchange of energy (an interaction) between Alice and Bob.

A preferred embodiment of the invention provides that the detectors ( DA1 ) and or ( DA2 ) and or ( DAL ) and or ( DB1 ) and or ( DB2 ) can be realized by means of superconducting detectors. Basically, the [ 23 ] Superconducting detectors described in detail.

A preferred embodiment of the invention provides that both Alice and Bob with the in shown structure can send or receive information. To do this, both only need to have the option of optionally using a beam splitter ( STB ) in your setup (if this is to serve as a receiver) or if it is to serve as a transmitter for the measurement MT0 a detector ( DSPG ) integrate into this or remove it when the measurement MT1 to be carried out.

A further preferred embodiment of the invention provides that the method for interaction-free information transmission described here is used to be able to transmit information that is secure against eavesdropping. The particular advantage in the method presented here can be seen in the fact that it is not only secure against eavesdropping, but there is also no possibility of recognizing the time at which information is transmitted. To do this, only the photon transit time ( P1 ) and ( P2 ) from the sources ( Q1 ) and ( Q2 ) up to the detectors used for the detection at Alice, so on the transit time of the photons ( P3 ) and ( P4 ) from the sources ( Q1 ) and ( Q2 ) be tuned up to the build up of Bob that at the time the photons ( P1 ) and ( P2 ) in Alice, the photons ( P3 ) and ( P4 ) are immediately in front of Bob's body, preferably only a few meters away from the body.

For the success of the procedure for the interaction-free transfer of information proposed here, it is of crucial importance how the wording "Knowledge that is basically available about the quantum system under consideration" is to be correctly interpreted. The generally accepted idea here is that it does not matter at what point in time the "principally accessible knowledge of a quantum system" is available. But what does that mean specifically? The experiment considered here raises the question, at which location (area) must the "principally accessible knowledge of a quantum system" be available (in the sense of "in principle can be called up in a classic way") and at what point in time? If knowledge is a classic property, knowledge must always be represented by energy. If Alice because of your measurement MT1 knows which polarization the photons ( P3 ) and ( P4 ) and the distance between Alice and Bob is sufficiently large, this knowledge of Alice (if it is a classic property) at the time when the photons ( P3 ) and ( P4 ) the point PB on the beam splitter ( STB ) do not even exist there in principle. The question also arises: If Alice at a certain point in time for a type of measurement ( MT0 or MT1 ) decides and implements this decision on your superstructure, does Bob have this knowledge immediately or only at a delayed time (due to the time it takes light to cover the distance between the superstructures)? If the experiment can nevertheless be successfully carried out, this shows that an interaction-free transfer of information can take place immediately and knowledge is not a classic property. The proposed procedure for the interaction-free transfer of information can thus also help to clarify very basic questions.

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QUOTES INCLUDE IN THE DESCRIPTION

This list of documents listed by the applicant has been generated automatically and is only included 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.

Patent literature cited

• DE 102010047168 [0117]
• US 9443200 B2 [0117]
• DE 102012000044 [0117]
• DE 102017005947 [0117]

Non-patent literature cited

• R.P. Feynman, R.B. Leighton and M.L. Sands in "The Feynman Lectures on Physics" (Addison-Wesley Publishing Co., Inc. Reading, 1989) [0117]
• DM. Greenberger, M.A. Horne, A. Zeilinger, Phys. Today, 22, (August 1993) [0117]
• A. Zeilinger, Am. J. Phys. 49, 882, (1981) [0117]
• W. Pauli, The Connection between Spin and Statistics; Phys. Rev. Vol. 58, 716-732 (1940). An alternative derivation of the relationship between spin and statistics can be found in: Arthur Jabs; Connecting spin and statistics in quantum mechanics; Foundations of Physiks 40 (7), 776-792, 793-794 (2010) [0117]
• H. de Riedmatten, et al; Quantum interference with photon pairs created in spatially separated sources; Physical Review A 67, 022301 (2003) [0117]
• A. Elitzur and L. Vaidman; Foundations of Physics, Vol. 23, No. 7, 987-997 (1993) [0117]
• Paul Kwiat, Harald Weinfurter, Thomas Herzog and Anton Zeilinger; Interaction-free measurement; Physical Review Letters, Volume 74, No. 24, 4763-4766 (1995) [0117]
• David Bohm; The implicit order. Basics of dynamic holism; 1st edition 1985, Dianus-Trikont Buchverlag GmbH, ISBN 3-88167-117-X [0117]
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• Markus Oberparleiter; Bosonian and Fermionic two-photon statistics at the beam splitter; Diploma thesis, University of Insbruck, 1997 [0117]

Claims (7)

Method for the interaction-free transmission of information by means of entangled photons, in which a source (Q1) simultaneously generates two photons (P1) and (P3) entangled in at least one secondary degree of freedom and a source (Q2) simultaneously produces two photons (P2) entangled in at least one secondary degree of freedom and (P4) and the transmitter of the information (Alice) for transmitting a binary coded information by means of a measurement of the first type (MT0) at the photons (P1) and (P2) that consisting of the photons (P3) and (P4) Converted the quantum system into a mixed state of the first type (Z1) and thereby encoding the first logical state, preferably the logical 0, and by means of a measurement of the second type (MT1) on the photons (P1) and (P2) that the photons ( P3) and (P4) existing quantum system is converted into a mixed state of the second type (Z2) and thereby encodes the second logical state, preferably the logical 1, and the The recipient of the information (Bob) subjects the state of the photons (P3) and (P4) to a state analysis with regard to the properties of the mixed state of the first type (Z1) and the mixed state of the second type (Z2) and from this the transmitted logic states for the purpose an interaction-free transmission of information can be recognized, characterized in that a symmetrical phase grating (SPG), preferably a symmetrical reflection grating, is used for the measurement of the first type (MT0) and is used in the pulse modes (A1) and (A0) on the symmetrical phase grating ( SPG) incoming photons (P1) and (P2) on the symmetrical phase grating (SPG) are diffracted into the pulse mode (AL). Procedure according to Claim 1 , characterized in that the degree of polarization freedom and / or the energetic degree of freedom is used as the secondary degree of freedom. Procedure according to one of the Claims 1 to 2nd , characterized in that a beam splitter, preferably a symmetrical beam splitter, is used for the state analysis of the photons (P3) and (P4). Procedure according to one of the Claims 1 to 3rd , characterized in that polarizing beam splitters and / or dichroic beam splitters are used for the measurements (MT0) and / or (MT1). Procedure according to one of the Claims 1 to 4th , characterized in that superconducting detectors are used to detect the photons used. Procedure according to one of the Claims 1 to 5 , characterized in that the transit time of the photons (P1) and (P2) from the sources (Q1) and (Q2) to the detectors used for the detection at Alice, so on the transit time of the photons (P3) and (P4) from the sources (Q1) and (Q2) to the structure of Bob that at the time when the photons (P1) and (P2) are detected in Alice, the photons (P3) and (P4) immediately before Bob is set up, preferably only a few meters away from the set-up, and thereby a tap-proof transmission of information is realized. Procedure according to one of the Claims 1 to 6 , characterized in that symmetrical phase grating (SPG) can be switched on and off electrically by means of a control unit.

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