Method for estimating displacement operator operating quantity using photon number state

Abstract

The application discloses a method for estimating displacement operator operation quantity by using photon number state, comprising the following steps: measuring average photon number contained in a photon number state contaminated by thermal noise, and estimating the thermal noise; using the photon number state contaminated by the thermal noise as a probe to detect an unknown displacement operator operation quantity, and detecting photon number of an output probe state; deducing a photon number probability distribution of the output probe state under the thermal noise; detecting the photon number of the output probe state for multiple times, and estimating the displacement operator operation quantity to be estimated according to the photon number probability distribution of the output probe state; the method is no longer limited to ideal conditions, and can realize estimation of the displacement operator operation quantity under the thermal noise; and the method can estimate the thermal noise mixed in a photon number state preparation process, and further improve the overall performance of estimation of the displacement operator operation quantity.

Description

Technical Field

[0001] This invention belongs to the field of quantum detection technology, specifically relating to a method for estimating shift operator operands using photon number states. Background Technology

[0002] Quantum detection-based estimation techniques are an important branch of quantum technology, used to estimate the magnitude of unknown operands in quantum operations. For example, Kennedy-type receivers are a type of quantum receiver capable of outperforming classical homodyne receivers. In Kennedy-type receivers, precise control of shift operator operands is required. This necessitates precise calibration of the system's shift operator operands using quantum detection-based estimation methods. Specifically, photon number states can be used to estimate the shift operator operands. This method, compared to coherent state-based estimation methods, has the advantage of not requiring a phase reference and offers superior estimation performance.

[0003] Current photon number state estimation methods have two limitations: First, they do not consider the contamination of the light source by thermal noise during the photon number state preparation process, which leads to a decrease in the overall performance and reliability of the estimation scheme. Second, in order to apply the photon number state-based estimation method to thermal noise environments, it is necessary to accurately estimate the thermal noise photon number to ensure the reliability of subsequent estimation of displacement operator operations.

[0004] Based on the aforementioned technical problems in the existing technology, this invention proposes a method for estimating the displacement operator operation quantity using photon number states. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of existing technologies by providing a method for estimating displacement operator operations using photon number states.

[0006] The present invention adopts the following technical solution:

[0007] A method for estimating displacement operator operands using photon number states includes:

[0008] The average number of photons contained in the photon number state contaminated by thermal noise is measured to estimate the environmental thermal noise.

[0009] Using photon number states contaminated by thermal noise as probes, unknown displacement operator operations are detected, and the number of photons in the output probe state is measured; the probability distribution of the number of photons in the output probe state under thermal noise is derived.

[0010] The number of photons in the output probe state is detected multiple times, and the displacement operator operation quantity to be estimated is estimated based on the probability distribution of the number of photons in the output probe state.

[0011] Furthermore, measuring the average number of photons contained in the photon number state contaminated by thermal noise includes:

[0012] The input vacuum state |0> is due to thermal noise n t The disturbance becomes a mixed state, and the density operator of the mixed state Represented as:

[0013]

[0014] Where α∈C, C is the entire complex field, n t Let |α> be the average number of photons generated by thermal excitation, and |α> be the coherent state. The coherent state |α> is represented by the Fock state as follows:

[0015]

[0016] Where |n> represents the Fokker state, and n represents the number of photons contained in the quantum state;

[0017] The density operator generates the operator k times. The quantum state obtained after the operation is:

[0018]

[0019] in, For annihilation operators, For the generation operator, k is the number of iterations of the generation operator, tr(·) is the trace of the operator, and the first moment of the number of photons X contained in the quantum state is:

[0020] E(X)=k(n t +1)+n t ……(4).

[0021] Furthermore, the estimation of environmental thermal noise includes:

[0022] Quantum state affected by thermal noise Perform M photon number measurements, and the resulting M photon numbers constitute a sample X = X1, X2, ..., X... M , x1, x2, ..., x m For the corresponding sample values, the population first moment E(X) is equal to the sample first moment:

[0023]

[0024] Therefore, the value of the parameter to be estimated is:

[0025]

[0026] Furthermore, before deriving the photon number probability distribution of the output probe state under thermal noise, we represent the coherent state of the shifted photon number state under thermal noise:

[0027] Photon number states Through unknown displacement operator operation Subsequently, the displacement photon number state under thermal noise is formed.

[0028]

[0029] in, For the shift operator, N k The trace of a quantum state after a shift operation is represented as:

[0030]

[0031] Arbitrary density operator Represented by coherent state notation, it is a function R(α) related to two coherent states |α> and |β>. * ,β), α,β∈C, where C is the entire complex field:

[0032]

[0033] Where |α> and |β> are two different coherent states, it can be seen from equation (9) that the displacement photon number state after thermal noise pollution Represented by coherent state notation as follows:

[0034]

[0035] Will With N k Substituting the expression into equation (11), we obtain the coherent state representation of the density operator as follows:

[0036]

[0037] Where, μ * It is the conjugate of the displacement offset μ. The photon number state for the displacement bias μ is represented as:

[0038]

[0039] Simplify equation (11):

[0040]

[0041] Here, f(n,m) is the simplified function expression containing n and m.

[0042] Furthermore, the derivation of the photon number probability distribution of the output probe state under thermal noise includes:

[0043] Displaced photon states under thermal noise are mixed states, and are represented as coherent states. With Fokk Two forms, two representations of the relation:

[0044]

[0045] The above equation (14) is used to obtain the result. The expression;

[0046] The distribution law p(n) of the photon number in the mixed state represented by the density operator is:

[0047]

[0048] When m = n, the distribution law of the number of photons at the receiver, p(n), is obtained as follows:

[0049]

[0050] Where i and j are integers not greater than k, the above formula holds when n≥i and n≥j, otherwise the probability is zero; |α> and |β> are coherent states, n and m are the number of photons corresponding to the number states |n> and |m>, n t N represents the average number of thermal photons. c The displacement operator operation quantity to be estimated. It is a generalized Laguerre polynomial.

[0051] Furthermore, estimating the displacement operator operation quantity to be estimated based on the photon number probability distribution of the output probe state includes:

[0052] The joint distribution law is obtained based on the number of photons detected in the output probe state sample.

[0053] The likelihood function of the sample is obtained from the joint distribution law of the sample values;

[0054] The displacement operator operation is estimated using the maximum likelihood estimation method.

[0055] Furthermore, the joint distribution law obtained from the photon number samples of the detected output probe states includes:

[0056] According to the maximum likelihood estimation method, the number of photons in the output probe state is detected and recorded M times at the detection end. The number of photons detected constitutes a sample N = N1, N2, ..., N M Based on the photon number probability distribution p(n) of the output probe state under thermal noise, the sample N = N1, N2, ..., N from the population is given. M The joint distribution law is:

[0057]

[0058] Furthermore, the likelihood function of the sample, obtained from the joint distribution law of the sample values, includes:

[0059] The number of photons n1, n2, ..., n of the output probe states was detected M times. M These correspond to samples N1, N2, ..., N M A sample value, sample N1, N2, ..., N M The observed values ​​n1, n2, ..., n are obtained. M The probability is:

[0060] P(N1=n1,N2=n2,...,N M =n M )……(18);

[0061] From equation (18), the likelihood function L(N) of sample N is obtained. c )for:

[0062]

[0063] Where, N c For the displacement operator to be estimated, the likelihood function L(N) is... c N represents the amount of operation to be estimated for the displacement operator. c The function.

[0064] Furthermore, based on the maximum likelihood estimation method, the estimation of the displacement operator operations includes:

[0065] Take samples n1, n2, ..., n M The shift operator operation that corresponds to the largest likelihood function is N. c The estimated value is then the estimated value of the displacement operator operation. Satisfy the following formula:

[0066]

[0067] The beneficial effects of this application are:

[0068] The method for estimating displacement operator operations using photon number states described in this application is applicable to detection estimation schemes under thermal noise backgrounds. Compared with existing estimation schemes based on photon number states, the method is no longer limited to ideal conditions and can achieve estimation of displacement operator operations under thermal noise conditions.

[0069] The method can estimate the magnitude of thermal noise mixed in during the photon number state preparation process, thereby improving the overall performance of estimating displacement operator operations. Attached Figure Description

[0070] Figure 1 This is a schematic diagram illustrating the principle of displacement operator operand estimation based on photon number state under thermal noise conditions in an embodiment of this application.

[0071] Figure 2Under the thermal noise conditions in the embodiments of this application, k = 5, n t =0.1,N c When = 0.1, the photon number distribution of the output probe is shown. Detailed Implementation

[0072] To better understand the above-mentioned objectives, features and advantages of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, unless otherwise specified, the embodiments and features in the embodiments of this application can be combined with each other.

[0073] To address the issue that existing photon number state estimation schemes do not consider thermal noise contamination, this application aims to provide a method for estimating displacement operator operations under thermal noise conditions using photon number states. Thermal noise is difficult to completely avoid during photon number state preparation and is a major source of noise. Therefore, the estimation method proposed in this application focuses on the impact of thermal noise on detection performance. First, the thermal noise is estimated using the method of moments. Then, based on the thermal noise estimation result, the maximum likelihood estimation method is further used to estimate the displacement operator operations, ultimately achieving accurate estimation of displacement operator operations under thermal noise conditions.

[0074] Example

[0075] The method for estimating displacement operator operands using photon number states includes:

[0076] Step 1: Considering the influence of thermal noise in the photon number state preparation process, measure the average number of photons contained in the photon number state contaminated by thermal noise, and estimate the environmental thermal noise through the moment estimation method;

[0077] Step 2: Using the photon number state contaminated by thermal noise as a probe, the unknown displacement operator operation quantity is detected, and the photon number of the output probe state is detected; then, based on the relationship between the coherent expression method and the Focke-Kiehl expression method of arbitrary density operators, the probability distribution of the photon number of the output probe state under thermal noise is derived.

[0078] Step 3: Detect the number of photons in the output probe state multiple times, and use the maximum likelihood estimation method to estimate the displacement operator operation quantity to be estimated based on the probability distribution of the number of photons in the output probe state.

[0079] Specifically, step 1 includes:

[0080] Step 1.1: Based on the properties of the vacuum thermal state, derive the expected number of photons contained in the photon number state contaminated by thermal noise;

[0081] The input vacuum state |0> is due to thermal noise n t The disturbance becomes a mixed state, and the density operator of this mixed state... It can be represented as:

[0082]

[0083] Where C is the entire complex field, n t Let |α> be the average number of photons generated by thermal excitation, and |α> be the coherent state. The coherent state |α> can be represented by the Fock state as follows:

[0084]

[0085] Where |n> represents the number state, i.e., the Fokker state, and n represents the number of photons contained in the quantum state;

[0086] The density operator generates the operator k times. The quantum state obtained after the operation is:

[0087]

[0088] in, For annihilation operators, For the generation operator, k is the number of iterations of the generation operator, tr(·) is the trace of the operator, and the first moment of the number of photons X contained in the quantum state is:

[0089] E(X)=k(n t +1)+n t ;

[0090] In this embodiment, thermal noise is considered as the main influencing factor, and the preparation process of the photon number state is as follows: Figure 1 The light source section is shown;

[0091] Step 1.2: Estimate the thermal noise incorporated into the light source part according to the moment estimation method;

[0092] Quantum state affected by thermal noise Perform M photon number measurements, and the resulting M photon numbers constitute a sample X = X1, X2, ..., X... M , x1, x2, ..., x m For the corresponding sample values, according to the principle of moments, the population first moment E(X) is equal to the sample first moment:

[0093]

[0094] Therefore, the value of the parameter to be estimated is:

[0095]

[0096] It should be noted that before estimating the displacement operator operations, the introduced thermal noise n needs to be pre-calculated. t Make an estimate.

[0097] Step 2 includes:

[0098] Step 2.1, coherent state representation of the shifted photon number state under thermal noise;

[0099] Photon number states Through unknown displacement operator operation Subsequently, the displacement photon number state under thermal noise is formed. Right now:

[0100]

[0101] in, For the shift operator, N c =|μ| 2 N represents the displacement operator operation to be estimated. k The trace of a quantum state after a shift operation is represented as:

[0102]

[0103] Arbitrary density operator It can be expressed by coherent state representation as a function R(α) related to two coherent states |α> and |β>. * ,β), α,β∈C, where C is the entire complex field:

[0104]

[0105] Here, |α> and |β> are two different coherent states, representing the displacement photon number states after thermal noise contamination. Represented by coherent state notation as follows:

[0106]

[0107] Will With N k Substituting the expression into the following equation, we obtain the coherent state representation of the density operator as follows:

[0108]

[0109] Where, μ * It is the conjugate of the displacement offset μ. For a photon number state with a displacement bias of μ, it is represented as:

[0110]

[0111] Perform R(α) * The simplification of ,β) gives:

[0112]

[0113] where f(n,m) is a simplified function expression containing n and m;

[0114] Step 2.2: Obtain the average photon number distribution law according to the relationship between the coherent state representation and the Fock state representation;

[0115] The displaced photon state under thermal noise is expressed in two forms: the coherent state and the Fock state , and the relational expression of the two representation forms is:

[0116]

[0117] The expression of is obtained from the above formula, and the photon number distribution law p(n) for the mixed state represented by the density operator is:

[0118]

[0119] It can be seen from the above formula that when m = n, the photon number distribution law p(n) at the receiving end is obtained, and the corresponding expression is as follows:

[0120]

[0121] where i and j are integers not greater than k, the above formula holds when n≥i and n≥j, otherwise the probability is zero, |α> and |β> are coherent states, n and m are the photon numbers corresponding to the number states |n> and |m>, n t is the average thermal photon number, is the generalized Laguerre polynomial;

[0122] It should be noted that the displaced photon state under thermal noise is a mixed state, which is expressed in two forms: coherent state <α|ρ|β> and Fock state <n|ρ|m;

[0123] As a specific embodiment, the photon number distribution diagram of the output probe when k=5, N c =0.1, n t =0.1 is given, as shown in Figure 2 .

[0124] Step 3 includes:

[0125] Step 3.1: Obtain the joint distribution law according to the photon number samples of the detected output probe state:

[0126] According to the principle of the maximum likelihood estimation method, the photon number of the output probe state is detected M times at the detection end and recorded, and the measured photon numbers constitute the sample N = N1, N2, ..., N M ; according to the derived photon number probability distribution p(n) of the output probe state under thermal noise, the sample N = N1, N2, ..., N from the population is given MThe joint distribution law is:

[0127]

[0128] Among them, the photon number distribution law p(n) of the probe state output by the detection end and the displacement operator operation quantity N to be estimated are... c Related;

[0129] Step 3.2: Obtain the likelihood function of the sample based on the joint distribution law of the sample values;

[0130] The number of photons N1,n2,...,n of the output probe states detected M times M It corresponds to samples N1, N2, ..., N M A sample value, sample N1, N2, ..., N M The observed values ​​n1, n2, ..., n are obtained. M The probability is:

[0131] P(N1=n1,N2=n2,...,N M =n M );

[0132] From the above formula, we obtain the likelihood function L(N) of sample N. c )for:

[0133]

[0134] Where, N c For the displacement operator to be estimated, the likelihood function L(N) is... c N represents the amount of operation to be estimated for the displacement operator. c The function;

[0135] Step 3.3: Estimate the displacement operator operation amount according to the maximum likelihood estimation method;

[0136] According to the principle of maximum likelihood estimation, the sample values ​​obtained correspond to the estimated value with the highest probability, i.e., the maximum value of the likelihood function. The sample values ​​n1, n2, ..., n are chosen such that the sample values ​​n1, n2, ..., n are the most probable values. M The shift operator operation that corresponds to the largest likelihood function is N. c The estimated value is then the estimated value of the displacement operator operation. Satisfy the following formula:

[0137]

[0138] This application is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of this application. Various changes and modifications can be made to this application without departing from the spirit and scope thereof, and all such changes and modifications fall within the scope of the claims. The scope of protection of this application is defined by the appended claims.

Claims

1. A method for estimating displacement operator operands using photon number states, characterized in that, include: The average number of photons in the photon number state contaminated by thermal noise is measured to estimate the environmental thermal noise. Before deriving the photon number probability distribution of the output probe state under thermal noise, the coherent state of the shifted photon number state under thermal noise is represented. Photon number states via unknown displacement operator operation Subsequently, the displacement photon number state under thermal noise is formed. : ……（7）； in, For shift operators, The trace of a quantum state after a shift operation is represented as: ……（8）； Arbitrary density operator Represented by coherent state notation as two coherent states and Related functions , For the entire complex field: ……（9）； in and As two different coherent states, it can be seen from equation (9) that the displacement photon number state after thermal noise contamination is... Represented by coherent state notation as follows: ……（10）； Will and Substituting the expression into equation (11), we obtain the coherent state representation of the density operator as follows: ……（11）； in, Displacement offset conjugate, The displacement offset is The photon number state is represented as: ……（12）； Simplify equation (11): ……（13）； in, To simplify, it contains and The function expression; Using photon number states contaminated by thermal noise as probes, unknown displacement operator operations are detected, and the photon count of the output probe state is measured. The probability distribution of the photon count of the output probe state under thermal noise is derived, including: Displaced photon states under thermal noise are mixed states, and are represented as coherent states. With Fokk Two forms, two representations of the relation: ……（14）； The above equation (14) is used to obtain the result. The expression; Determining the distribution law of photon number in mixed states represented by the density operator for: ……（15）； when The distribution law of the number of photons at the receiving end is obtained. The expression is as follows: ……（16）； in, and Not greater than integers, the above formula in and If it holds true, then the probability is zero; and It is a coherent state. and Number of states and The corresponding number of photons, The average number of thermal photons, The displacement operator operation quantity to be estimated. It is a generalized Laguerre polynomial; The number of photons in the output probe state is detected multiple times, and the operation quantity of the displacement operator to be estimated is estimated based on the probability distribution of the number of photons in the output probe state.

2. The method for estimating displacement operator operations using photon number states according to claim 1, characterized in that, Measuring the average number of photons contained in a photon number state contaminated by thermal noise includes: Input vacuum state Due to thermal noise The disturbance becomes a mixed state, and the density operator of the mixed state Represented as: ……（1）； in, For the entire complex field, The average number of photons excited by thermal excitation. It is a coherent state, a coherent state The Fokker state is represented as: ……（2）； in, For the Fokker state, This represents the number of photons contained in a quantum state; density operator after Secondary generation operator The quantum state obtained after the operation is: ……（3）； in, For annihilation operators, To generate operators, The number of iterations for generating the operator. For the trace of an operator, the number of photons contained in a quantum state. The first moment is: ……（4）。 3. The method for estimating displacement operator operations using photon number states according to claim 1 or 2, characterized in that, The estimation of environmental thermal noise includes: Quantum state affected by thermal noise conduct The photon number measurement yielded The number of photons constitutes the sample. , For the corresponding sample values, the population first moment Equal to the first moment of the sample: ……（5）； Therefore, the value of the parameter to be estimated is: ……（6）。 4. The method for estimating displacement operator operations using photon number states according to claim 1, characterized in that, The estimation of the displacement operator operation quantity to be estimated based on the photon number probability distribution of the output probe state includes: The joint distribution law is obtained based on the number of photons detected in the output probe state sample. The likelihood function of the sample is obtained from the joint distribution law of the sample values; The displacement operator operation is estimated using the maximum likelihood estimation method.

5. The method for estimating displacement operator operations using photon number states according to claim 4, characterized in that, The joint distribution law obtained from the photon count samples of the detected output probe state includes: Based on the maximum likelihood estimation method, detection is performed at the detection end. The number of photons in the next output probe state is recorded, and the measured number of photons constitutes the sample. Based on the photon number probability distribution of the output probe state under thermal noise Provide a sample from the population. The joint distribution law is:

6. The method for estimating displacement operator operations using photon number states according to claim 5, characterized in that, The likelihood function of a sample, derived from the joint probability distribution of the sample values, includes: Detection Number of photons in the next output probe state The corresponding sample A sample value, sample Observations were obtained The probability is: ……（18）； The sample can be obtained from equation (18). likelihood function for: ……（19）； in, For the displacement operator to be estimated, the likelihood function is... The amount of operation for the displacement operator to be estimated The function.

7. The method for estimating displacement operator operations using photon number states according to claim 6, characterized in that, The estimation of the displacement operator operations based on the maximum likelihood estimation method includes: Take sample The displacement operator operation corresponding to the maximum likelihood function is used as The estimated value is then the estimated value of the displacement operator operation. Satisfy the following formula: ……（20）。

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