Configuration acquisition method and system for realizing maximum violation five-setting CHSH measurement device

Abstract

The invention discloses a configuration acquisition method and system for realizing a maximum violation five-setting CHSH measurement device, and relates to the technical field of quantum information processing, and the method comprises the steps: S1, obtaining the probability amplitude of the pure state of a pair of two-dimensional non-maximum entangled particle A and particle B, and calculating the concurrency; s2, constructing an incidence matrix based on the concurrency, and mapping the measurement vector of the particle B from the Bloch spherical surface to the ellipsoid surface by using the incidence matrix; s3, taking five points on the same plane on the ellipsoid as vertexes of a five-pointed star, wherein a perimeter formula of the five-pointed star is equivalent to a correlation function maximum value formula; s4, traversing vertex positions of the five-pointed star to obtain a maximum perimeter value; s5, obtaining the measurement vector of the particle B and the measurement vector of the particle A to obtain the optimal measurement configuration; and S6, configuring a quantum measurement device to measure the particles A and B. According to the method, the problem of setting the maximum value of the CHSH correlation function is converted into the problem of solving the maximum value of the perimeter of the ellipsoid inscribed five-pointed star, and the optimal measurement scheme is obtained through simple numerical fitting.

Description

Technical Field

[0001] This invention relates to the field of quantum information processing technology, and in particular to a method and system for acquiring the configuration of a five-set CHSH measurement device to achieve maximum violation. Background Technology

[0002] Nonlocality is one of the key characteristics that distinguishes quantum mechanics from classical physics, and its core manifestation is the violation of Bell's inequality. Under local hidden variable theory, the behavior of a system should satisfy Bell's inequality; however, quantum mechanics can violate this inequality under certain conditions, reflecting the existence of nonclassical correlations in quantum systems.

[0003] The most commonly used Bell inequality is the Clauser-Horne-Shimony-Holt (CHSH) inequality, which is applicable to two-particle, two-measurement, and two-output scenarios. To improve the robustness of Bell inequalities in experiments, researchers have developed multi-set extensions, such as the five-set CHSH inequality, which has shown stronger performance in tasks such as quantum randomness authentication, self-test protocols, and device-independent quantum key distribution.

[0004] However, in practical quantum systems, due to limitations such as entanglement state preparation errors and channel decoherence, it is often difficult to obtain the ideal maximally entangled state. In most cases, only non-maximally entangled states can be achieved. Therefore, how to achieve the maximum violation of the five-set CHSH inequality under this condition has become a key problem that urgently needs to be solved. Existing literature has given the maximum violation of the five-set CHSH inequality under the condition of maximizing entanglement, but for any non-maximally entangled state, how to effectively find a measurement scheme that maximizes the violation of the five-set CHSH inequality still lacks a unified and efficient theoretical method.

[0005] The experimental measurement process for setting the CHSH inequality five times is described as follows: The expression for the CHSH inequality five times (correlation function) is given:

[0006] (1);

[0007] If the experiment measures If the value is greater than 10, it proves the nonlocality of the system. In the actual experimental verification, the expected function in equation (1) It can be obtained through coincidence counting, and is expressed as:

[0008] (2);

[0009] coincidence count To project particle A onto the state Simultaneously projecting the B particle onto the state The joint measurement count. Among them, the two-dimensional particle state , It can be represented as:

[0010] (3). Summary of the Invention

[0011] To address the above problems, this invention proposes a method for obtaining the configuration of a five-set CHSH measurement device that achieves maximum violation. Through geometric mapping, the problem of finding the maximum value of the five-set CHSH correlation function is transformed into the problem of finding the maximum circumference of a pentagram inscribed in an ellipsoid. Thus, the optimal measurement scheme can be obtained through simple numerical fitting, thereby achieving maximum violation of the five-set CHSH inequality.

[0012] On the one hand, the configuration acquisition method for the five-set CHSH measurement device to achieve the maximum violation is as follows:

[0013] S1, obtain the probability amplitude of the pure state of a pair of two particles A and B that are not maximally entangled in two dimensions, and calculate the concurrency degree of quantization of the degree of entanglement based on the probability amplitude.

[0014] S2, construct the correlation matrix based on the concurrency, and use the correlation matrix to map the measurement vector of particle B from the Bloch sphere to an ellipsoid;

[0015] S3, taking five points on the same plane on the ellipsoid as the vertices of the pentagram, the formula for the perimeter of the pentagram formed is equivalent to the formula for the maximum value of the CHSH inequality correlation function of particles A and B.

[0016] S4: Traverse and search the positions of the five points among the vertices of the pentagram to find the maximum value of the pentagram's perimeter, and obtain the maximum value of the corresponding correlation function formula;

[0017] S5, set the Bloch spherical vector corresponding to the vector pointing from the center of the ellipsoid to the vertex of the pentagram as the measurement vector of particle B, and set the unit vector corresponding to the edge of the pentagram as the measurement vector of particle A, thus completing the optimal measurement configuration of setting the CHSH inequality.

[0018] S6. According to the optimal measurement configuration of the CHSH inequality set in the fifth step, a quantum measurement device is set up to perform joint projection measurement on particle A and particle B to obtain the measurement value used to verify the quantum violation behavior of the CHSH inequality.

[0019] Preferably, the method for calculating the concurrency of the degree of entanglement based on the probability amplitude is as follows:

[0020] ;

[0021] in, Indicates the degree of concurrency. ∈(0,1]; Indicates that particle A is in The state and particle B are in The probability amplitude of the state.

[0022] Preferably, the correlation matrix is ​​a diagonal matrix; the diagonal elements of the diagonal matrix are respectively , and -1; where, Indicates the degree of concurrency.

[0023] Preferably, the semi-axial lengths of the ellipsoid are respectively , And 1.

[0024] Preferably, the five-set CHSH inequality correlation function for particles A and B is expressed as:

[0025] ;

[0026] ;

[0027] in, , , , and This represents the measurement vectors of five selected points on the ellipsoid. Indicates an association function; Indicates the definition symbol; , , , and This represents the measurement vectors of the five selected points on the ellipsoid in the Bloch sphere of particle B; , , , and This represents the corresponding measurement vector in the Bloch sphere of particle A; This represents an association function.

[0028] Preferably, the unit vector corresponding to the edge of the pentagram is represented as:

[0029] , , , , ;

[0030] in, This indicates taking the modulus.

[0031] Preferably, the formula for the maximum value of the CHSH inequality correlation function is expressed as follows:

[0032] ;

[0033] in, This indicates the maximum value of the correlation function; This indicates taking the modulus.

[0034] Preferably, the position of the pentagram corresponding to the maximum circumference of the pentagram is: the pentagram is located on the ellipse formed by the intersection of the ellipsoid and a plane containing the Z-axis; and the major axis of the ellipsoid is located on the Z-axis.

[0035] Preferably, in S4, the vertex position of the pentagram with Z-axis symmetry is selected to reduce the number of parameters required when traversing and searching for the vertex position of the pentagram.

[0036] On the other hand, the configuration acquisition system for the five-set CHSH measurement device that achieves maximum violation includes the following:

[0037] The concurrency acquisition module is used to acquire the probability amplitude of the pure state of a pair of two particles, A and B, which are not maximally entangled in two dimensions, and to calculate the concurrency degree of quantification of the degree of entanglement based on the probability amplitude.

[0038] The correlation matrix acquisition and mapping module is used to construct the correlation matrix based on the concurrency and use the correlation matrix to map the measurement vector of particle B from the Bloch sphere to an ellipsoid.

[0039] The pentagram construction module is used to construct a pentagram with five points on the same plane on the ellipsoid as vertices. The formula for the perimeter of the pentagram is equivalent to the formula for the maximum value of the CHSH inequality correlation function of particles A and B.

[0040] The traversal module is used to traverse and search the positions of the five points among the vertices of the pentagram, find the maximum perimeter of the pentagram, and obtain the maximum value using the corresponding correlation function maximum value formula;

[0041] The measurement configuration acquisition module is used to set the Bloch spherical vector corresponding to the vector pointing from the center of the ellipsoid to the vertex of the pentagram as the measurement vector of particle B, and the unit vector corresponding to the edge of the pentagram as the measurement vector of particle A, thus completing the optimal measurement configuration of setting the CHSH inequality.

[0042] The measurement module is used to set up a quantum measurement device according to the optimal measurement configuration of the five CHSH inequalities, and to perform joint projection measurements on particles A and B to obtain measurement values ​​for verifying the quantum violation behavior of the CHSH inequalities.

[0043] Compared with the prior art, the present invention has the following beneficial effects:

[0044] This invention transforms the problem of finding the maximum value of the five-set CHSH correlation function for a pair of two-dimensional non-maximally entangled particles into the problem of finding the maximum circumference of a pentagram inscribed in an ellipsoid. Thus, the optimal measurement scheme can be obtained through simple numerical fitting, achieving the maximum violation of the five-set CHSH inequality. This method has the advantages of clear structure, no need for full traversal, and ease of simulation, providing a new design idea and analysis tool for Bell inequality experiments in non-maximally entangled states. Attached Figure Description

[0045] The present invention will now be described in further detail with reference to the accompanying drawings;

[0046] Figure 1 This is a flowchart illustrating the method for obtaining the configuration of the CHSH measurement device for achieving the maximum violation according to an embodiment of the present invention;

[0047] Figure 2 This is a schematic diagram illustrating the geometric model of the correlation function of the configuration acquisition method for the five-set CHSH measurement device to achieve the maximum violation in an embodiment of the present invention; wherein, (a) represents a schematic diagram of the Bloch sphere of two particles; (b) represents a schematic diagram of the correlation matrix K acting on the Bloch sphere of particle B.

[0048] Figure 3 This is a schematic diagram of the circumference of the inscribed pentagon of the ellipse after mapping, representing the configuration acquisition method of the CHSH measurement device for achieving the maximum violation according to an embodiment of the present invention.

[0049] Figure 4 This invention provides an experimental measurement device suitable for the photon orbital angular momentum degree of freedom, which is based on the method for obtaining the configuration of a five-position CHSH measurement device to achieve the maximum violation.

[0050] Figure 5 The diagram below shows the structural block diagram of the configuration acquisition system for the CHSH measurement device to achieve the maximum violation in an embodiment of the present invention. Detailed Implementation

[0051] The present invention will be further described below through specific embodiments.

[0052] like Figure 1 As shown, the configuration acquisition method for the five-set CHSH measurement device to achieve the maximum violation is as follows:

[0053] Let a pair of concurrency degrees be Two-particle pure states ∈ (0,1] =0 indicates a non-entangled state. =1 indicates maximizing the entanglement state. In this embodiment, the particle is a photon, and its wavefunction can be written as:

[0054] (4);

[0055] The incidence matrix corresponding to this state is a diagonal matrix. Measurement state On the Bloch sphere, it is represented as a unit vector from the center of the sphere to the surface of the sphere. ,like Figure 2 As shown in (a). Under this mapping, the expectation function Expressed as measurement vector Inner product with the incidence matrix K: Then equation (1) can be expressed as:

[0056] (5);

[0057] To solve for the maximum correlation function value, this embodiment introduces a direction from the center of the sphere to the semi-major axis as... New measurement vector of the ellipsoid and 1: ,like Figure 2 As shown in (b), the correlation function can be further expressed as:

[0058] (6);

[0059] when , , , , At that time, the correlation function Take the maximum value, expressed as:

[0060] (7);

[0061] at this time, Equivalent to five points on the ellipsoid ( The perimeter of the five-pointed star formed by ( ) is as follows: Figure 2 As shown in (b). According to the literature [Advanced Photonics Research, 2025, : 2500105], the optimal measurement vector lies in the same plane containing the Z-axis, and a set of optimal measurement vectors with Z-axis symmetry can be selected. Therefore, the largest pentagram is located between the ellipsoid and this plane ( On the intersection of the ellipse (face), such as Figure 3 As shown in (a).

[0062] To reduce the complexity and parameter requirements of numerical simulation, this embodiment adopts... Noodles (order) for A set of optimal measurement basis vectors with Z-axis symmetry is used to reduce the parameters of the required measurement basis vectors from five vectors to two, and numerical solutions are performed based on this configuration. Figure 3 As shown in (b), point ,point , and points Symmetric about the Z-axis, satisfying ,but:

[0063] (8);

[0064] In the middle, set ,but , , , .exist In the middle, =0, then we can get , , , According to equation (8), we can obtain:

[0065] (9);

[0066] Command Point and Traverse the entire ellipse and output... Maximum value and corresponding point and By using the coordinates, we can obtain the maximum violation of the CHSH inequality and the optimal measurement scheme for the five settings.

[0067] In summary, the core of this embodiment lies in transforming the problem of setting the maximum value of the CHSH correlation function into a problem where the semi-major axis is... This paper addresses the problem of finding the maximum perimeter of a pentagram inscribed in an ellipsoid of 1. The proposed method offers advantages such as clear structure, no need for full traversal, and ease of simulation, providing a new design approach and analytical tool for Bell's inequality experiments in non-maximally entangled states.

[0068] The following section, with specific mathematical expressions and steps, further explains the five-set CHSH inequality maximum violation measurement method based on ellipsoidal geometry optimization in this embodiment.

[0069] Construction of concurrency degree and correlation matrix for non-maximally entangled states:

[0070] This embodiment of the measurement is applicable to two-particle pure states. Taking photons as an example, the two-photon orbital angular momentum entangled state prepared through a spontaneous parametric downconversion process can be expressed as follows: Selecting a subspace The two-dimensional entangled state is represented as follows:

[0071] (10);

[0072] parameter and They are not equal, but satisfy the normalization condition: For the measurement of orbital angular momentum entangled states, such as... Figure 4 As shown, a spatial light modulator (SLM) is typically used to convert the target state to a fundamental Gaussian beam, which is then coupled into a single-mode fiber. For other modes, since they cannot be converted to a fundamental Gaussian beam by the SLM, they cannot enter the single-mode fiber, thus preventing projection measurement of specific modes. By introducing photon A and photon B into the measurement module and performing coincidence measurement on the signals output from the two single-mode fibers, the parameters can be obtained. and Specifically, the parameters and This is determined through the following relationship: ,in, This indicates that the projection of photon A onto the state was measured. Simultaneously, the B particle is projected onto the state. The coincidence count; This indicates that the projection of particle A onto the state is measured. Simultaneously, the B particle is projected onto the state. The concurrency count is calculated according to equation (4).

[0073] (11);

[0074] The incidence matrix K is defined in diagonal form: This matrix compresses the Bloch sphere into a principal axis. The ellipsoid represents the measurement direction space of the B particle.

[0075] Traversal search Maximum measured value: for parameter and A scan is performed, with the scan range being [0, 2π]. The Bell parameter can be obtained for each scan according to equation (9). Compare the results obtained from each scan. The optimal solution is the combination of measurement matrices that maximizes the parameter among all candidate solutions.

[0076] For example, assume the experimental system measures a count per second. According to equation (11), the concurrency can be calculated as follows: 0.7, construct the corresponding entangled state, represented as:

[0077] (12);

[0078] Point , obtained by parameterized search The maximum value is 8.5709, and the corresponding measurement vector is the optimal measurement scheme. The specific measurement vector is as follows: By mapping the measurement vector back to the orbital angular momentum measurement state, the optimal measurement state can be obtained as follows:

[0079] (13);

[0080] Based on the above measurement states, the experimenter can directly perform the following: Figure 4 The measurement setup is configured on the spatial light modulator, and joint projection measurements are performed on particles A and B. The expected values ​​of each term of the correlation function are statistically calculated and summed to finally verify the quantum violation behavior of the inequality.

[0081] like Figure 5 As shown, the present invention also discloses a configuration acquisition system for a five-set CHSH measurement device that achieves maximum violation, comprising:

[0082] The concurrency acquisition module 501 is used to acquire the probability amplitude of the pure state of a pair of two particles, A and B, which are not maximally entangled in two dimensions, and to calculate the concurrency degree of quantification of the degree of entanglement based on the probability amplitude.

[0083] The correlation matrix acquisition and mapping module 502 is used to construct the correlation matrix based on the concurrency and use the correlation matrix to map the measurement vector of particle B from the Bloch sphere to an ellipsoid.

[0084] The pentagram construction module 503 is used to construct a pentagram with five points on the same plane on the ellipsoid as vertices. The formula for the perimeter of the pentagram is equivalent to the formula for the maximum value of the CHSH inequality correlation function of particles A and B.

[0085] Traversal module 504 is used to traverse and search the positions of the five points among the vertices of the pentagram, find the maximum perimeter of the pentagram, and obtain the maximum value of the corresponding correlation function formula.

[0086] The measurement configuration acquisition module 505 is used to set the Bloch spherical vector corresponding to the vector pointing from the center of the ellipsoid to the vertex of the pentagram as the measurement vector of particle B, and set the unit vector corresponding to the edge of the pentagram as the measurement vector of particle A, thus completing the optimal measurement configuration of setting the CHSH inequality.

[0087] Measurement module 506 is used to set up a quantum measurement device according to the optimal measurement configuration of the CHSH inequality set in the five settings, and to perform joint projection measurement on particle A and particle B to obtain measurement values ​​for verifying the quantum violation behavior of the CHSH inequality.

[0088] The specific implementation of the configuration acquisition system for the five-set CHSH measurement device that achieves maximum violation is the same as the configuration acquisition method for the five-set CHSH measurement device that achieves maximum violation, and will not be repeated in this embodiment.

[0089] The above are merely specific embodiments of the present invention, but the design concept of the present invention is not limited thereto. Any non-substantial modifications made to the present invention using this concept shall be considered as infringing upon the protection scope of the present invention.

Claims

1. A method for obtaining the configuration of a five-set CHSH measurement device to achieve maximum violation, characterized in that, Includes the following steps: S1, obtain the probability amplitude of the pure state of a pair of two particles A and B that are not maximally entangled in two dimensions, and calculate the concurrency degree of quantization of the degree of entanglement based on the probability amplitude. S2, construct the correlation matrix based on the concurrency, and use the correlation matrix to map the measurement vector of particle B from the Bloch sphere to an ellipsoid; S3, taking five points on the same plane on the ellipsoid as the vertices of the pentagram, the formula for the perimeter of the pentagram formed is equivalent to the formula for the maximum value of the CHSH inequality correlation function of particles A and B. S4: Traverse and search the positions of the five points among the vertices of the pentagram to find the maximum value of the pentagram's perimeter, and obtain the maximum value of the corresponding correlation function formula; S5, set the Bloch spherical vector corresponding to the vector pointing from the center of the ellipsoid to the vertex of the pentagram as the measurement vector of particle B, and set the unit vector corresponding to the edge of the pentagram as the measurement vector of particle A, thus completing the optimal measurement configuration of setting the CHSH inequality. S6. According to the optimal measurement configuration of the CHSH inequality set in the fifth step, a quantum measurement device is set up to perform joint projection measurement on particle A and particle B to obtain the measurement value used to verify the quantum violation behavior of the CHSH inequality.

2. The method for obtaining the configuration of the five-set CHSH measurement device for achieving maximum violation according to claim 1, characterized in that, The concurrency degree for calculating the quantization of entanglement based on probability amplitude is as follows: ； in, Indicates the degree of concurrency. ∈(0,1]; Indicates that particle A is in The state and particle B are in The probability amplitude of the state.

3. The method for obtaining the configuration of the five-set CHSH measurement device for achieving maximum violation according to claim 1, characterized in that, The correlation matrix is ​​a diagonal matrix; the diagonal elements of the diagonal matrix are respectively , and -1; where, Indicates the degree of concurrency.

4. The method for obtaining the configuration of the five-set CHSH measurement device for achieving maximum violation according to claim 3, characterized in that, The semi-axial lengths of the ellipsoid are respectively , And 1.

5. The method for obtaining the configuration of the five-set CHSH measurement device for achieving maximum violation according to claim 1, characterized in that, The five-set CHSH inequality correlation function for particles A and B is expressed as: ； ； in, , , , and This represents the measurement vectors of five selected points on the ellipsoid. Indicates an association function; Indicates the definition symbol; , , , and This represents the measurement vectors of the five selected points on the ellipsoid in the Bloch sphere of particle B; , , , and This represents the corresponding measurement vector in the Bloch sphere of particle A; This represents an association function.

6. The method for obtaining the configuration of the five-set CHSH measurement device for achieving maximum violation according to claim 5, characterized in that, The unit vector corresponding to the edge of the pentagram is represented as: ， ， ， ， ； in, This indicates taking the modulus.

7. The method for obtaining the configuration of the five-set CHSH measurement device for achieving maximum violation according to claim 5, characterized in that, The formula for the maximum value of the CHSH inequality correlation function is expressed as follows: ； in, This indicates the maximum value of the correlation function; This indicates taking the modulus.

8. The method for obtaining the configuration of the five-set CHSH measurement device for achieving maximum violation according to claim 1, characterized in that, The position of the pentagram corresponding to the maximum perimeter of the pentagram is as follows: the pentagram is located on the ellipse formed by the intersection of the ellipsoid and a plane containing the Z-axis; and the major axis of the ellipsoid is located on the Z-axis.

9. The method for obtaining the configuration of the five-set CHSH measurement device for achieving maximum violation according to claim 8, characterized in that, In S4, the vertex positions of the pentagram with Z-axis symmetry are selected to reduce the number of parameters required when traversing and searching for the vertex positions of the pentagram.

10. A configuration acquisition system for a five-position CHSH measurement device that achieves maximum violation, characterized in that, include: The concurrency acquisition module is used to acquire the probability amplitude of the pure state of a pair of two particles, A and B, which are not maximally entangled in two dimensions, and to calculate the concurrency degree of quantification of the degree of entanglement based on the probability amplitude. The correlation matrix acquisition and mapping module is used to construct the correlation matrix based on the concurrency and use the correlation matrix to map the measurement vector of particle B from the Bloch sphere to an ellipsoid. The pentagram construction module is used to construct a pentagram with five points on the same plane on the ellipsoid as vertices. The formula for the perimeter of the pentagram is equivalent to the formula for the maximum value of the CHSH inequality correlation function of particles A and B. The traversal module is used to traverse and search the positions of the five points among the vertices of the pentagram, find the maximum perimeter of the pentagram, and obtain the maximum value using the corresponding correlation function maximum value formula; The measurement configuration acquisition module is used to set the Bloch spherical vector corresponding to the vector pointing from the center of the ellipsoid to the vertex of the pentagram as the measurement vector of particle B, and the unit vector corresponding to the edge of the pentagram as the measurement vector of particle A, thus completing the optimal measurement configuration of setting the CHSH inequality. The measurement module is used to set up a quantum measurement device according to the optimal measurement configuration of the five CHSH inequalities, and to perform joint projection measurements on particles A and B to obtain measurement values ​​for verifying the quantum violation behavior of the CHSH inequalities.

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