Spatial quantum communication channel compensation method based on positive semidefinite programming

Abstract

The invention provides a space quantum communication channel compensation method based on positive semidefinite programming. The method comprises the steps that a sender carries out initial coding on modulated photons; the receiver carries out initial decoding on the received photons; the receiver detects the photons to obtain a Kraus operator describing a noise channel; based on the noise channel Kraus operator, the initial coding operator and the initial decoding operator, the sender performs iterative optimization on the coding operator to obtain the optimal solution of the coding operator, so that the composite channel is optimal; based on the optimal solution of the channel Kraus operator and the optimal solution of the decoding operator, the receiver performs iterative optimization on the decoding operator to obtain the optimal solution of the decoding operator, so that the composite channel is optimal; and after the optimal solution of the decoding operator is obtained, the optimal solution is returned to a sender, iteration is repeated until the composite channel is optimal, and the photon state fidelity is calculated. According to the method, the channel is compensated by using the positive semi-definite programming, the channel is not required, and noise channels which do not meet the requirement of determining an error correction scheme can be processed.

Description

Technical Field

[0001] The invention belongs to the technical field of space quantum communication, and in particular relates to a space quantum communication channel compensation method based on semi-positive definite programming. Background Art

[0002] Quantum communication is based on the basic principles of quantum mechanics and implements the communication process in a way different from classical communication. It mainly includes branches such as quantum key distribution, quantum secure direct communication, and quantum secret sharing. Its security is guaranteed by the Heisenberg uncertainty principle and the no-cloning principle. Any eavesdropping will be detected by both communicating parties and it has physical unconditional security. In contrast, the security of classical communication depends on the computational complexity of certain mathematical problems (such as factorization of large prime numbers). In the future, with the continuous improvement of computing power, especially the vigorous development of research on quantum computers, its principle cannot guarantee security. Therefore, quantum information has huge development potential and application prospects, and may trigger many scientific and technological revolutions.

[0003] Quantum communication usually uses photons as communication carriers. On the one hand, this is because photons hardly decoherent during free space or fiber transmission, and the development level of linear optical components used to manipulate quantum bits is already very mature. On the other hand, the technical accumulation of classical optical communication has also facilitated the development of photon-based quantum communication. For example, fiber-optic communication technology has played an important role in quantum communication. Similar to classical laser communication, quantum communication channels also include optical fiber and free space. At present, optical fiber-based quantum key distribution has successfully achieved a communication distance of up to 509 km and is moving towards commercial application. However, problems such as birefringence, loss, and imperfect detection equipment and light sources in single-mode optical fibers limit the transmission distance of photons in optical fibers. Compared with optical fiber channels, the loss and decoherence effects in free space are weaker. Satellite-based quantum communication has achieved a communication distance of thousands of kilometers, laying the foundation for building a global quantum communication network.

[0004] Due to the limitation of the quantum no-cloning law (i.e., it is impossible to perfectly clone an unknown quantum state), quantum signal amplification cannot be achieved during quantum communication. Therefore, how to reduce the loss of quantum signals during transmission becomes the key to improving the transmission distance and speed of quantum communication.

[0005] The present invention proposes a spatial quantum communication channel compensation method based on semi-positive definite programming. The purpose of existing quantum error correction coding is to achieve perfect recovery of the quantum state to the initial state after passing through a noisy channel. The problem brought about by this is that the proposed error correction scheme requires that the channel itself must meet certain conditions, and the error correction effect may not be achieved for channels that do not meet the conditions. The method proposed in the present invention reduces the entire process to a semi-positive definite programming problem, and the corresponding optimization problem can be iterated between encoding and decoding. Since the solution process is iterated for a given noise form, an important advantage of this method is that there is no requirement for the channel itself, and noisy channels that do not meet the requirements of a perfect error correction scheme can be processed. At the same time, it can also ensure that the optimal solution within the feasible domain is obtained. Summary of the invention

[0006] The technical problem to be solved by the present invention is: in view of the shortcoming that the existing algorithm cannot effectively track the scene with complex acceleration changes, an interactive multi-model filtering algorithm based on an improved extended acceleration observer is proposed to realize the relative state estimation of space targets under complex maneuvers.

[0007] In order to solve the above technical problems, the technical solution adopted by the present invention is: a relative state estimation method under complex motion of a space target, comprising the following steps:

[0008] S1: The communication process is modulated on the polarization / phase / orbital angular momentum of the photon, and then enters the standard error-corrected quantum communication process, including the encoding operation C, the noise channel E, and the decoding operation R, that is, where ρ S is the density matrix of the initial photon state (the density matrix can be directly calculated from the photon state), ρ C is the density matrix encoding the photon state, σ C is the density matrix of the coded state affected by the noise channel, ρ′ S is the density matrix of the final photon state after the decoding operation;

[0009] S2: The rate of the quantum communication process is related to the fidelity of the initial photon state and the final photon state, denoted by F(ρ S , ρ′ S ), the greater the fidelity, the closer the final photon state is to the initial photon state, the smaller the noise effect of the transmission process is, and the higher the quantum communication rate is. The fidelity is defined for the two density matrices as:

[0010]

[0011] S3: The purpose of the whole system is to find the appropriate C and R to maximize the fidelity of the initial and final photon states, so it can be expressed as Where H is the Hilbert space where the initial state is located, and finding the minimum value of the state in H is to ensure the error correction fidelity in the worst case;

[0012] S4: The above optimization problem of fidelity is converted into: by designing the encoding operation C and the decoding operation R, for a given noisy channel E, the composite operation REC may be close to the unit matrix (i.e., the ideal case without noise, denoted as I). The similarity between two quantum operations is usually measured by channel fidelity. The channel fidelity of the composite mapping REC and the unit matrix can be expressed as:

[0013]

[0014] Where 0≤f≤1, n s The dimension of the density matrix representing the initial photon state, R r 、E e , C c are the operators and representations of the coding operation C, the noise channel E, and the decoding operation R, Tr represents the trace of the matrix, and |*| represents the absolute value;

[0015] S5: According to the Knill-Laflamme theorem, if and only if there exists a constant δ rec Satisfy R r E e C c =δ rec I and ∑ r，e，c |δ rec | 2 =1, f = 1, at this time the composite mapping REC is equivalent to the identity matrix I, that is, the composite channel R·E·C is equivalent to the ideal channel I;

[0016] S6: Convert the above channel fidelity into channel distance That is, d = 0 is equivalent to f = 1. represents the Frobenius norm

[0017] S7: The encoding operation, the noisy channel, the decoding operation and the operator are represented by {C c}, {E e}, {R r} are rewritten into Choi representation, denoted as C, E, and R, and then δ rec Arranged into a matrix Δ≡[δ rec ], so the channel distance d can be rewritten as From the orthogonality of the matrix, the optimization problem under channel fidelity can be obtained as:

[0018]

[0019] S8: First, optimize R, randomly initialize R and C, substitute E, R, C and the above optimization problem into the semi-positive definite programming solver toolbox, and solve the following optimization problem about Δ to get the optimal solution of R:

[0020]

[0021] From Δ, we can get where υ1 and u1 are Substitute the left and right singular value vectors in the singular value decomposition of R into the above formula again and solve until d1 no longer decreases;

[0022] S9: Then optimize C, substitute Δ, R and the following optimization problem about C into the semi-positive definite programming solver toolbox, and the optimal solution of C can be obtained:

[0023]

[0024] After obtaining the value of C, return to S8 and repeat the iteration until d2 no longer decreases;

[0025] S10: Based on the minimum value of d obtained in claim S9, the fidelity of the photon state can be calculated as f = (1-d min / 2n S ) 2 , where n S is the dimension of the system space;

[0026] S11: The key rate K in quantum communication process = Q*[1-H2(δ z )-H2(δ x )], where Q represents the photon reception rate, H2(x) = -xlog2x-(1-x)log2(1-x) represents binary entropy, δ z and δ x They respectively represent the error rates under the corresponding measurement basis. The above variables can all be obtained from the photon fidelity. Based on the photon state fidelity obtained in S10, the key rate of the quantum communication process can be calculated.

[0027] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are:

[0028] The advantages of the present invention are mainly that it provides a channel compensation method for the space quantum communication process, which can effectively reduce the influence of the noise channel on the single photon as the quantum communication carrier, and improve the key rate of the quantum communication process. Furthermore, the channel compensation method provided by the present invention has the following advantages: it can perform adaptive optimization for different channels, obtain the optimal encoding and decoding scheme, reduce the interference caused by channel noise, and improve the communication rate in the quantum communication process; the semi-positive programming method used has a relatively mature theory and solution tool, which is convenient for practical application; it has different principles from other channel compensation methods for space quantum communication, such as adaptive optics, and is convenient for multiplexing, further improving the key rate of the quantum communication process. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] In order to more clearly illustrate the technical solution of the embodiment of the present invention, the following is a brief introduction to the drawings required for describing the embodiment, in which:

[0030] Figure 1 It is the standard process of quantum communication including encoding and decoding;

[0031] Figure 2 The fidelity comparison between this method and three-bit redundant coding in a bit flip noise channel is shown, and the relationship between the fidelity and the bit flip probability under standard 3-bit coding and this method is shown. DETAILED DESCRIPTION

[0032] The technical solution of the present invention will be clearly and completely described below in conjunction with the embodiments. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0033] The first embodiment of the present invention provides a spatial quantum communication channel compensation method based on semi-positive definite programming. The standard quantum communication process including encoding and decoding in a noisy environment is as follows: Figure 1 As shown, in quantum communication, the quantum state carrying information 0 is usually recorded as |0>, and the quantum state carrying information 1 is recorded as |1>. Bit flipping is a common type of noise, which flips 0 to 1 and 1 to 0 with a certain probability. The corresponding operator form is The probability of flipping is P. The common three-bit redundant coding method is to use three identical bits 0(1) to represent 0(1). The bit value is determined by the majority of the final state. If |011> is received, it is judged as 1, and if |001> is received, it is judged as 0. At this time, the operator form of the noise channel contains 8 Kraus operators, namely:

[0034]

[0035] A1 and A2 represent the situations where no flipping occurs and flipping occurs, respectively.

[0036]

[0037] After obtaining the operator form of the noise channel, substitute it into the following pseudo code process:

[0038] Algorithm steps:

[0039] Step 1: Initialize R and C

[0040] Step 2: Repeat the following process

[0041] (1) Optimizing R-repeats

[0042] (a) Solve the optimization equation for Δ

[0043]

[0044] (b) R can be obtained from the following relationship: where υ1 and u1 are The left and right singular value vectors in the singular value decomposition of ;

[0045] (c) until d1 no longer decreases;

[0046] (2) Optimize C

[0047] (a) Solve the optimization formula for C:

[0048]

[0049] (3) Until d2 no longer decreases.

[0050] The above optimization problems can be solved by the YALMIP toolbox. YALMIP is a toolbox based on MATLAB. Its programming language is the same as MATLAB, so it has the advantage of being easy to perform various matrix operations. It supports a variety of optimization problems, such as linear optimization, quadratic optimization, semi-positive definite optimization, etc. After using the YALMIP toolbox, you only need to set the objective function of the optimization problem and the corresponding constraints. The subsequent transformation and solution of the problem are automatically completed by the toolbox.

[0051] Figure 2The relationship between the standard 3-bit coding and the fidelity obtained by this method in such a noisy channel and p is given. It can be seen that when p≤0.5, the standard coding and the results obtained by this method are the same, and when p>0.5, this method can obtain higher fidelity. This shows that deterministic quantum coding has certain requirements for channel noise and is not suitable for environments that do not meet the requirements. This method can be improved according to different forms of noise and has stronger robustness to noise interference environments.

[0052] The preferred embodiments of the present invention disclosed above are only used to help explain the present invention. The preferred embodiments do not describe all the details in detail, nor do they limit the invention to specific implementation methods. Obviously, many modifications and changes can be made according to the content of this specification. This specification selects and specifically describes these embodiments in order to better explain the principles and practical applications of the present invention, so that those skilled in the art can understand and use the present invention well. The present invention is limited only by the claims and their full scope and equivalents.

Claims

1. A spatial quantum communication channel compensation method based on semi-positive definite programming, characterized in that: A digital compensation method based on semi-positive programming is provided for the noise disturbance caused by the spatial channel to the photons. The method includes the following steps: encoding and decoding initialization, channel operator generation, encoding operator iterative optimization, decoding operator iterative optimization, and fidelity calculation. The method includes the following steps: S1: The communication process is modulated on the polarization / phase / orbital angular momentum of the photon, and then enters the standard error-corrected quantum communication process, including the encoding operation C, the noise channel E, and the decoding operation R, that is, where ρ S is the density matrix of the initial photon state (the density matrix can be directly calculated from the photon state), ρ c is the density matrix encoding the photon state, σ c is the density matrix of the coded state affected by the noise channel, ρ′ s is the density matrix of the final photon state after the decoding operation; S2: The rate of the quantum communication process is related to the fidelity of the initial photon state and the final photon state, denoted by F(ρ S , ρ′ S ), the greater the fidelity, the closer the final photon state is to the initial photon state, the smaller the noise effect of the transmission process is, and the higher the quantum communication rate is. The fidelity is defined for the two density matrices as: S3: The purpose of the whole system is to find the appropriate C and R to maximize the fidelity of the initial and final photon states, so it can be expressed as Where H is the Hilbert space where the initial state is located, and finding the minimum value of the state in H is to ensure the error correction fidelity in the worst case; S4: The above optimization problem of fidelity is converted into: by designing the encoding operation C and the decoding operation R, for a given noisy channel E, the composite operation REC may be close to the unit matrix (i.e., the ideal case without noise, denoted as I). The similarity between two quantum operations is usually measured by channel fidelity. The channel fidelity of the composite mapping REC and the unit matrix I can be expressed as: Where 0≤f≤1, n s The dimension of the density matrix representing the initial photon state, R r 、E e , C c are the operators and representations of the coding operation C, the noise channel E, and the decoding operation R, Tr represents the trace of the matrix, and |*| represents the absolute value; S5: According to the Knill-Laflamme theorem, if and only if there exists a constant δ rec Satisfy R r E e C c =δ rec I and ∑ r，e，c |δ rec | 2 =1, f = 1, at this time the composite mapping REC is equivalent to the identity matrix I, that is, the composite channel R·E·C is equivalent to the ideal channel I; S6: Convert the above channel fidelity into channel distance That is, d = 0 is equivalent to f = 1. represents the Frobenius norm S7: The encoding operation, the noisy channel, the decoding operation and the operator are represented by {C c }, {E e }, {R r } are rewritten into Choi representation, denoted as C, E, and R, and then δ rec Arranged into a matrix Δ≡[δ rec ], so the channel distance d can be rewritten as From the orthogonality of the matrix, the optimization problem under channel fidelity can be obtained as: S8: First, optimize R, randomly initialize R and C, substitute E, R, C and the above optimization problem into the semi-positive definite programming solver toolbox, and solve the following optimization problem about Δ to get the optimal solution of R: From Δ, we can get where υ1 and u1 are Substitute the left and right singular value vectors in the singular value decomposition of R into the above equation again and solve until d1 no longer decreases; S9: Then optimize C, substitute Δ, R and the following optimization problem about C into the semi-positive definite programming solver toolbox, and the optimal solution of C can be obtained: After obtaining the value of C, return to S8 and repeat the iteration until d2 no longer decreases; S10: Based on the minimum value of d obtained in claim S9, the fidelity of the photon state can be calculated as f = (1-d min / 2n S ) 2 , where n S is the dimension of the system space; S11: The key rate K in quantum communication process = Q*[1-H2(δ z )-H2(δ x )], where Q represents the photon reception rate, H2(x) = -xlog2x-(1-x)log2(1-x) represents binary entropy, δ z and δ x They respectively represent the error rates under the corresponding measurement basis. The above variables can all be obtained from the photon fidelity. Based on the photon state fidelity obtained in S10, the key rate of the quantum communication process can be calculated.

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