CN114978350A - Noise-free linear amplification method for polarization-time segment super-coding FOCK state - Google Patents
Noise-free linear amplification method for polarization-time segment super-coding FOCK state Download PDFInfo
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Abstract
The invention provides a noise-free linear amplification method for a polarization-time segment supercoded FOCK state, which can perform noise-free linear amplification on the FOCK state of any polarization-time segment double-freedom-degree supercoded by constructing a multi-mode parallel quantum scissors scheme and reserve the coding characteristics of photons on polarization and time segment freedom degrees; firstly, a sender passes a group of FOCK states through an N-order beam splitter to uniformly divide incident states into N paths, and each path is provided with a group of same quantum scissors devices and auxiliary photons are prepared; and the incident state on each path enters the quantum scissors device, and when the quantum scissors devices on all paths successfully operate, all output photons are converged by an N-order beam splitter in an arrangement mode opposite to that of the previous N-order beam splitter to obtain an output FOCK state. By adjusting the transmissivity of the variable beam splitter in each quantum shear device, the average photon number of the FOCK state can be effectively increased.
Description
Technical Field
The invention belongs to the technical field of quantum communication, and mainly relates to a noise-free linear amplification method of FOCK state of polarization-time segment two-degree-of-freedom coding.
Background
Quantum communication means that the transmission of information is realized to the rationale of utilizing quantum mechanics, and compared with classical communication, quantum communication has better security. Photons are the best carriers for quantum communication. The existing quantum communication protocol mainly carries out coding in the polarization or space mode freedom degree of a single photon. In fact, in addition to the above two modes, degrees of freedom such as time slice, frequency, etc. can be used for encoding. Meanwhile, coding in multiple degrees of freedom of photons can effectively improve the channel capacity of the photons, and the coding is widely used in various quantum communication schemes to increase the practical communication efficiency. When photons are transmitted in an actual channel, due to the existence of environmental noise in the channel, photon transmission loss inevitably exists, so that the photons are exponentially attenuated along with the increase of the channel length during transmission, and the serious image affects the communication length of quantum communication. Quantum noise-free linear amplification is an important approach to combat photon transmission loss. In 2009, Ralph and Lund put forward the concept of quantum noise-free linear amplification for the first time; in 2010, Gisin proposed a noiseless linear amplification scheme based on linear optics and applied this scheme to device independent quantum key distribution. Most of the currently proposed noiseless linear amplification protocols aim at amplification of single photon states or two-photon entangled states of discrete variables, and can keep the coded information of the photons in the degrees of freedom such as polarization, time slice, frequency and the like unchanged while realizing amplification, but the noiseless linear amplification protocols aim at continuous variables. The FOCK state is a common continuous variable entanglement state, is an important resource of quantum communication, and is widely applied to various quantum communication protocols at present. In 2020, a noiseless linear amplification protocol for a FOCK state is proposed, however, the noiseless amplification protocol does not consider the encoding situation of photons, and therefore, the coded information of photons cannot be retained while achieving FOCK state amplification. Therefore, the research on the noiseless amplification protocol in the FOCK state of the multi-freedom coding has important value for realizing the efficient quantum communication of the FOCK state based on the multi-freedom coding.
Disclosure of Invention
In order to solve the technical problem of noise-free linear amplification of FOCK state of simultaneous multi-free coding, the invention considers that the polarization and time slice freedom degree of each photon in FOCK state are coded simultaneously, and designs a noise-free linear amplification method of FOCK state of polarization-time slice two-freedom-degree coding by introducing the idea of parallel quantum scissors, thereby improving the average photon number of FOCK state and effectively protecting the coding condition of photons on polarization and time slice self-freedom degree.
The invention relates to a noise-free linear amplification method of a polarization-time segment super-coding FOCK state, which comprises the following steps:
step 1, a sender prepares a group of arbitrarily polarized-time-sliced double-freedom-degree supercoded FOCK states, and the FOCK states pass through a first N-order beam splitter array, and the first N-order beam splitter array evenly divides incident states into N paths, so that quantum states on each path are attenuated into weak coherent states;
step 2, installing the same quantum scissors device on each path, and providing 4 auxiliary polarized photons in each path; the incident state and the auxiliary state of each path enter the quantum scissors device, if the quantum scissors devices on some paths do not obtain the successful detector response condition, all output states are discarded, and the scheme is terminated; if the quantum scissors devices on all paths obtain successful detector response conditions, the output states of all the quantum scissors devices are reserved, and step 3 is carried out;
step 3, inputting the output states of the quantum scissors devices on the N paths into a second N-order beam splitter array, and converging the output states of the N paths into a total path for output;
and 4, measuring the photon number of all output ports of the second N-order beam splitter array except the first output port, and when the photon number of the output ports of other paths is zero, the noise-free linear amplification scheme of the polarization-time slice super coding FOCK state is successful.
Further, in step 1, firstly, an operator is defined:
is a single mode operator at a particular frequency, where m ∈ (1,2, …, infinity), γ m To satisfy
The photon state generated by the single photon source under the current experimental condition is the following FOCK state:
considering | α> A Is coherent, so it can be changed into
In the form of (1), wherein
A probability coefficient for each term; two more operators are defined:
is an operator of a single mode on a horizontal polarization,
is the operator of a single mode on horizontal polarization, tau 0 ,τ 1 Respectively, the normalized complex weight coefficients;
is a single-mode operator over a long time span,
is a single-mode operator over a short range of time, ξ 0 ,ξ 1 Respectively, normalized complex weight coefficients thereof. Obtaining the polarization-time of the sender preparationThe form of FOCK state of fragment two-degree-of-freedom super coding is as follows:
wherein
|H>And | V>Respectively defined as horizontal polarization and vertical polarization, alpha and beta are respectively normalized complex weight coefficients thereof; i S>And | L>Defined as short and long time runs, respectively, and δ, η are their normalized complex weight coefficients, respectively. Coefficient in two degrees of freedom satisfies | α ∞ 2 +|β| 2 ＝1，|δ| 2 +|η| 2 1, where all four coefficients are complex.
Further, the FOCK state is split into N paths after passing through the beam splitter, and the weak coherent state obtained on each path is in the form of:
wherein
C 0 Probability coefficient of vacuum state, C 1 Is the probability coefficient of a single photon.
Further, in step 2, the same quantum scissors device is installed on each path, and 4 auxiliary polarized photons are provided in each path, wherein the quantum state is | H > | H > | V > | V >; the quantum shear device comprises 10 polarization beam splitters, 2 variable beam splitters with the transmittance of t, 2 variable beam splitters and 50 variable beam splitters: 50 beam splitters, 4 pockets and 8 single photon detectors.
Further, in each path, the incident state and the auxiliary state enter the quantum shear device simultaneously, and the operation process is as follows: the incident state is divided into two paths after passing through the PBS, and a passenger box PC is respectively arranged on the two paths S 、PC L Wherein PC S The polarization state of photons in the time-short mode S can be reversed, as can PC L Can be turned over for a long timeThe polarization state of the photon on mode L; two auxiliary photons | H are respectively input into the two auxiliary paths>And | V>By setting the path length in the auxiliary path, an accurate correspondence | H of the time slice and the polarization is achieved S >And | V L >(ii) a A variable beam splitter VBS with a transmission ratio t is arranged in each of the two auxiliary paths, the auxiliary state reflected by the VBS enters the output path, and the transmitted auxiliary state and the incident state pass through 50: 50 beam splitter BS, and the eight detectors are divided into D 1 D 3 、D 2 D 4 、D 5 D 7 、D 6 D 8 Four groups, and defined in the four groups of detectors, a single quantum scissor arrangement is successfully amplified when only one detector per group accurately detects one photon; if the quantum scissors devices on part of paths do not obtain the successful response condition of the detector, discarding all output states; if all quantum scissor devices have a successful detector response, all output states are retained.
Further, for a single quantum scissor device, when the input is in a vacuum state, the scheme success probability is t 4 (ii) a When the input is a single photon, the probability of success of the scheme is t 3 (1-t); thus, the probability of success P of a quantum scissors arrangement on a single path 0 ＝C 0 t 4 +C 1 t 3 (1-t) wherein C 0 t 4 Is the amplified success probability when the input is in a vacuum state, C 1 t 3 (1-t) is the amplification success probability when the input is a single photon.
Further, when the input state is a single photon state, if a quantum scissors device obtains a successful detector response condition, the obtained output state is:
when the input state is a vacuum state, if a quantum scissors device obtains a successful detector response condition, the obtained output state is also a vacuum state.
Further, in step 3, the total output state obtained after the output states of the N paths are merged by the N-order beam splitter is in the form of:
wherein the amplification factor is:
coefficient of
Further, in step 4, the amplification factor of the total output state is:
the work probability of the scheme assembly is as follows:
when t is less than 0.5, g is obtained s And > 1, the average number of photons in the output state is greater than the average number of photons in the input state, thereby realizing the amplification of the FOCK state.
The invention has the beneficial effects that: according to the invention, by constructing a multi-mode parallel quantum scissors scheme and adjusting the transmissivity of the variable beam splitter in each quantum scissors device, the average photon number of an output FOCK state can be effectively improved, noiseless linear amplification is carried out on the FOCK state of double-degree-of-freedom super-coding of any polarization-time segment, and the coding characteristics of photons on the polarization and time segment freedom degrees are reserved; the protocol only uses optical equipment commonly used under the current experimental conditions, can be realized under the existing experimental conditions, and has strong applicability.
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FIG. 1 is a schematic flow diagram of the present invention;
FIG. 2 is a schematic diagram of the inventive noiseless linear amplification scheme for multiple modes;
FIG. 3 is a schematic diagram of a single quantum scissors on a path in the multimode parallel scheme of the present invention;
FIG. 4 is a schematic diagram of successful detector response in a single quantum scissor arrangement; wherein ● represents the corresponding detector accurately detecting a photon.
Detailed Description
In order that the present invention may be more readily and clearly understood, there now follows a more particular description of the invention in terms of specific embodiments and reference to the accompanying drawings.
Because the photon can generate transmission loss in the process of transmitting information in an imperfect channel, the pure state and the mixed state of the photon are mixed with a certain probability, the efficiency of quantum communication is greatly reduced, the safety of the quantum communication is threatened, the loss of photon transmission can be effectively resisted by the noise-free linear amplification technology, and the noise-free linear amplification technology plays an important role in remote quantum communication. Aiming at the technical problems that the noise-free linear amplification protocol of continuous variables (such as coherent states) at the present stage is less, and the existing noise-free linear amplification scheme aiming at FOCK states cannot keep the coding information of FOCK states on the degrees of freedom of polarization and time slices, the invention provides a noise-free linear amplification method of coherent states (FOCK states) of polarization-time slice double-degree-of-freedom super coding, which can perfectly keep the coding information of photons on the degrees of freedom of polarization and time slices while amplifying the FOCK states.
First we define the operator:
in the formula
Is a single mode operator at a particular frequency, where m ∈ (1,2, …, infinity), γ m To satisfy
Normalized complex weight coefficients of (a).
The photon state generated by a single photon source under the current experimental conditions is the following FOCK state:
now, the photon in FOCK state is encoded in two degrees of freedom of polarization and time slice, and the encoding form is as follows:
the formula satisfies | α uti 2 +|β| 2 ＝1，|δ| 2 +|η| 2 1, where the four coefficients of the two degrees of freedom are all complex.
Similar we define two operators again:
wherein
Is a single-mode operator on the horizontal polarization,
is a single-mode operator on the horizontal polarization,
is a single-mode operator over a long time span,
is a single mode operator over a short range of time.
The coded FOCK state form is as follows:
since quantum scissors can only be used for amplification of weak coherent states, i.e. | ψ> in ＝|0>+α|1>Thus, the amount ofThe sub-scissor scheme cannot be used directly for | α> A I have introduced a multi-mode parallelization approach.
First we build a multi-modal parallel framework. In FIG. 2, N-Splitter is an N-order beam Splitter array and Quantum Scissor is Quantum scissors. The scheme needs two N-order beam splitter arrays which are arranged in the opposite mode, N-1 vacuum assist devices, N quantum scissors devices and N-1 photon storage devices.
A bundle of input states | α> A The vacuum state generated by the N-1 vacuum assist devices and the first N-order beam splitter are input together, and the N-order beam splitter evenly divides the input state into N paths. | α>After passing through the N-stage beam splitter, the coherent state of each lane becomes | α'>I.e. by
Wherein
The input state of each path is obtained as:
since N is large, therefore
Very small, approaching 0, hence, | α'>The higher order terms of (2) can be omitted, so we only consider the first two terms; thus, the input state of a single path can be obtained
Wherein
The input state on one path is
Then, theWe consider the amplification case in a single quantum scissors device. In fig. 3, the PBS is a polarization beam splitter, the BS is 50: 50 beam splitters, VBS a variable beam splitter with transmission t, PC a podcast box, D i (i ═ 1,2 … …, 8) is a photon detector.
Assume the input state is:
assume the output state is:
incident state
From a to a 1 End input through PBS 1 Enter a 2 、a 3 To obtain an input state
In the figure, the PBS is a polarizing beam splitter that enables incident | H>Complete transmission, | V>And is completely refracted.
Input state | psi 2 >At a 2 、a 3 Respectively pass through the PC of the general box S 、PC L Obtaining an input state
Wherein the PC S Can be turned overPolarization state of photons in the time-short mode S, also PC L The polarization state of photons in the time-long mode L can be reversed, e.g. | H S >Through PC S Then will turn over to | V S >。
The scheme requires the use of up to four auxiliary polarized photons | H>|H>|V>|V>. At b 1 End and c 1 End simultaneous input of two auxiliary photons | H>And | V>；b 1 Two auxiliary photons at the end can smoothly pass through the PBS 2 、PBS 3 ，c 1 Two auxiliary photons at the end can also pass through the PBS smoothly 6 、PBS 7 . In PBS 2 、PBS 3 ，PBS 6 、PBS 7 The path lengths are set in the two paths so as to realize accurate correspondence of the time slice and the polarization. At | V>Is provided with a longer route to realize | V>And | L>In the correspondence of | H>Is provided with a shorter route to realize | H>And | S>To (c). Therefore we are in b 2 、c 2 Can obtain auxiliary state | psi 1 >。
In each of the two auxiliary paths is arranged a variable beam splitter VBS having a transmission of t at b 2 End, c 2 The auxiliary photons of the terminals respectively pass through VBS 1 、VBS 2 Obtaining an auxiliary state:
at this time, VBS is passed 1 、VBS 2 Reflected auxiliary state | ψ 2 >Respectively enter the output paths b 4 、c 4 A terminal; through the VBS 1 、VBS 2 Transmissive auxiliary state | ψ 2 >And input state
Respectively from b 3 a 4 、c 3 a 5 End-entry BS 1 、BS 2 . BS is a 50: and 50, when a single photon is input at two input ends of the BS, the two single photons can only be output from the same output end according to the bunching effect, and the two single photons are collectively represented as:
thus, we know the input state from the bunching effect
Through BS 1 、BS 2 Become later
Likewise, auxiliary state | ψ 2 >Through BS 1 、BS 2 Then becomes | psi 3 >：
Since the photon detector can accurately detect the number of incident photons,therefore we divide the eight detectors into D 1 D 3 、D 2 D 4 、D 5 D 7 、D 6 D 8 Four groups of detectors are defined, and when only one detector in each group accurately detects one photon, the quantum scissors device operates successfully; when other conditions occur, the quantum shear device fails to operate. And if the quantum scissors scheme on part of paths fails to operate, discarding all output states and terminating the scheme. And if all the quantum scissors devices obtain the successful detector response condition, the output states of all the quantum scissors devices are kept, and the next step is carried out. In fig. 4, a successful detector response is shown in a single quantum shear device. Wherein ● denotes that the corresponding detector detects exactly one photon, e.g. the first ● denotes D 1 、D 2 、D 5 、D 6 The four detectors detect exactly one photon. Thus we can calculate that a total of 16 cases can make the solution successful.
The input state is not input with photons, i.e. the input photon is 0, the input state is vacuum state, and the auxiliary photon is normally from b 1 、 c 1 And (4) end input. At this time, the following partial events can make the detector respond successfully to make the scheme successful, and we set the partial events as | phi 0 >：
Thus, even when no photons are input at the input, there is a probability that the detector will respond successfully due to the effect of the auxiliary photons, such that the scheme is successful, the amplification scheme success probability being t 4 . In this scheme, b is the same as 4 、c 4 No photon input at the end, so no photons enter the PBS 10 Therefore, the total output state is a vacuum state.
When the input is
When the auxiliary photon is normally from b 1 、 c 1 End input, only the following part of events can make the detector response successfully to make the scheme successful, and we set the part of events as | phi 1 >，
Because the | alpha ∞ is satisfied 2 +|β| 2 ＝1，|δ| 2 +|η| 2 The probability of success of the amplification scheme can be calculated as (1-t) t ═ 1 3 At this time, in b 4 、c 4 Obtaining an output state
The total output state can be obtained by comparing the above formula
Therefore, the encoding condition of the output state on two degrees of freedom is perfectly reserved, and the successful amplification condition is met.
The success probability of an amplification scheme of quantum scissors is as follows:
P＝C 1 t 3 (1-t)+C 0 t 4
output state
The fidelity of (1) is:
when F is present 1 >C 0 The amplification scheme may be considered successful. Fidelity F of output mixed state 1 Independent of the coefficients α, β, δ, η, and only with the fidelity C of the input mixture 0 And the transmittance t of VBS.
We have previously calculated the probability of success (1-t) t for the scheme when the input is a single photon 3 The success probability of the scheme when the input is in a vacuum state is t 4 . Because of | α'>The higher order terms of (2) may be omitted so i consider only the first two terms. It is thus possible to obtain:
the success probability of the single path at this time is:
order to
Substitution into P 1 The method comprises the following steps:
wherein
Setting output state
Wherein
Then, normalization processing is carried out on the output state to obtain:
at this time, the amplification factor
And the output states of the N paths after quantum scissors processing are converged and output on the second N-order beam splitter array in an arrangement mode opposite to that of the first N-order beam splitter array. When the photon numbers in the quantum memories of all the output ports of the second N-order beam splitter except the first output port are zero, the amplification scheme is successful.
In that
Under the conditions of (a), we can obtain the form of the total output state:
the scheme assembly work probability is as follows:
wherein
When t is less than 0.5, g s If the number of the photons is more than 1, the average number of the photons in the output state is larger than that in the input state, so that the FOCK state is amplified. As N approaches infinity, the scheme approaches the ideal quantum scissors amplification scheme, but the probability of success for this scheme decreases as N increases. The noise-free linear amplification method for the polar-time slice supercode FOCK state is now over.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the present invention further, and all equivalent variations made by using the contents of the present specification and the drawings are within the protection scope of the present invention.
Claims (9)
1. A noise-free linear amplification method for a polarization-time slice super-coding FOCK state is characterized by comprising the following steps:
step 1, a sender prepares a group of arbitrarily polarized-time-sliced double-freedom-degree supercoded FOCK states, and the FOCK states pass through a first N-order beam splitter, and the first N-order beam splitter evenly divides incident states into N paths, so that quantum states on each path are attenuated into weak coherent states;
step 2, installing the same quantum scissors device on each path, and providing 4 auxiliary polarized photons in each path; the incident state and the auxiliary state of each path enter the quantum scissors device, if the quantum scissors devices on some paths do not obtain the successful detector response condition, all output states are discarded, and the scheme is terminated; if all the quantum scissors devices obtain successful detector response conditions, output states of all the quantum scissors devices are reserved, and the step 3 is carried out;
step 3, inputting the output states of the quantum scissors devices on the N paths into a second N-order beam splitter array, and converging the output states of the N paths into a total path for output;
and 4, measuring the photon number of all output ports of the second N-order beam splitter except the first output port, and when the photon number of other output ports is zero, successfully performing the noise-free linear amplification scheme of the polarization-time slice supercoding FOCK state.
2. The method for noiseless linear amplification of the polar-time slice supercoded FOCK state according to claim 1, characterized in that in step 1, the operator is first defined:
is a single mode operator at a particular frequency, where m ∈ (1,2, …, infinity), γ m To satisfy
The obtained photon state generated by the single photon source under the current experimental condition is the following FOCK state:
considering | α> A Is coherent, so it can be changed into
In the form of (1), wherein
A probability coefficient for each term; two more operators are defined:
is a single-mode operator on the horizontal polarization,
is the operator of a single mode on horizontal polarization, tau 0 ,τ 1 Respectively, the normalized complex weight coefficients;
is a single-mode operator over a long time span,
is a single-mode operator over a short range of time, ξ 0 ,ξ 1 Respectively, normalized complex weight coefficients thereof. The form of the FOCK state of the polarization-time fragment double-freedom-degree super-coding prepared by the sender is as follows:
wherein
|H>And | V>Respectively defined as horizontal polarization and vertical polarization, alpha and beta are respectively normalized complex weight coefficients thereof; i S>And | L>Respectively defined as a short time range and a long time range, and delta and eta are respectively normalized complex weight coefficients thereof. Coefficient in two degrees of freedom satisfies | α ∞ 2 +|β| 2 ＝1，|δ| 2 +|η| 2 Where all four coefficients are complex numbers.
3. The noise-free linear amplification method for the FOCK state with the polarization-time slice supercoding function as claimed in claim 2, wherein the FOCK state is divided into N paths after passing through the first N-order beam splitter array, and weak coherent states obtained on each path are in the form of:
wherein
C 0 Is the probability coefficient in the vacuum state, C 1 Is the probability coefficient of a single photon.
4. The noise-free linear amplification method for the polarization-time segment supercoding FOCK state according to claim 1, wherein in step 2, the same quantum scissors device is installed on each path, and 4 auxiliary polarization photons are provided in each path, the quantum state of which is | H > | V >; the quantum shear device comprises 10 polarization beam splitters, 2 variable beam splitters with the transmittance of t, 2 variable beam splitters and 50 quantum shear devices, wherein the polarization beam splitters are arranged in the quantum shear device and are respectively in the following positions: 50 beam splitters, 4 pockets and 8 single photon detectors.
5. The noise-free linear amplification method for the FOCK state of polar-time slice supercoding of claim 4, wherein in each path, the incident state and the auxiliary state enter the quantum-scissor device simultaneously, and the operation process is as follows: the incident state is divided into two paths after passing through the PBS, and a passenger box PC is respectively arranged on the two paths S 、PC L Wherein PC S The polarization state of photons in the short-time mode S can be reversed, as can PC L The polarization state of photons in the time-long mode L can be reversed; two auxiliary photons | H are respectively input into the two auxiliary paths>And | V>By setting the path length in the auxiliary path, an accurate correspondence | H of the time slice and the polarization is achieved S >And | V L >(ii) a A variable beam splitter VBS with a transmission ratio t is arranged in each of the two auxiliary paths, the auxiliary state reflected by the VBS enters the output path, and the transmitted auxiliary state and the incident state pass through 50: 50 beam splitter BS, and the eight detectors are divided into D 1 D 3 、D 2 D 4 、D 5 D 7 、D 6 D 8 The four groups are defined in the four groups of detectors, and when only one detector in each group accurately detects one photon, the amplification of the single quantum shear device is successful; if the quantum scissors devices on partial paths do not obtain successful detector response conditions, discarding all output states; if all quantum scissor devices obtain successful detector response conditions, all output states are retained。
6. The method of claim 1, wherein for a single quantum scissor device, when the input is in a vacuum state, the scheme success probability is t 4 (ii) a When the input is a single photon, the scheme success probability is t 3 (1-t); thus, the probability of success P of a quantum-scissor arrangement on a single path 0 ＝C 0 t 4 +C 1 t 3 (1-t) wherein C 0 t 4 Is the amplified success probability, C, when the input is in a vacuum state 1 t 3 (1-t) is the amplification success probability when the input is a single photon.
7. The noise-free linear amplification method for the poled-time slice supercoded FOCK states of claim 6, wherein when the input states are single photon states, if a quantum-scissor device obtains a successful detector response, the resulting output states are:
when the input state is a vacuum state, if a quantum scissors device obtains a successful detector response condition, the obtained output state is also a vacuum state.
8. The noise-free linear amplification method for the FOCK state of polarization-time slice supercoding as claimed in claim 1, wherein in step 3, the output states of N paths are merged by an N-order beam splitter to obtain a total output state in the form of:
wherein the amplification factor is:
coefficient of performance
9. The noise-free linear amplification method for the FOCK state of polarization-time slice super coding as claimed in claim 1, wherein in the step 4, the amplification factor of the total output state is:
the scheme assembly work probability is as follows:
when t is less than 0.5, g is obtained s And > 1, the average number of photons in the output state is greater than the average number of photons in the input state, thereby realizing the amplification of the FOCK state.
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