Random true entanglement quantum state generation method based on PPT criterion

Abstract

The invention provides a random true entanglement quantum state generation method based on a PPT criterion, belongs to the field of quantum information, and aims to solve the problem of low efficiency in a traditional quantum state generation process. The method specifically comprises the steps that a pre-screening step is introduced when a completely random density matrix is generated, a minimum characteristic value is calculated through partial transposition of a subsystem or a partition, and a negative threshold value is set. And when all the minimum characteristic values are smaller than the threshold value, complex entanglement judgment is further carried out. And gradually reducing the threshold value to optimize the efficiency of generating the true entanglement state, and finally finding the optimal threshold value. According to the method, the calculation amount is remarkably reduced, the efficiency of generating the true entanglement quantum state is improved, and particularly, good expansibility is shown in a multi-volume subsystem. Compared with a traditional direct generation method, the generation efficiency can be improved by 71.67% at most by introducing the threshold value. According to the method, a more efficient true entanglement sample generation means can be provided for machine learning, and the method has wide application value and expansibility.

Description

Technical Field

[0001] The present invention relates to the field of quantum information, and specifically to a random true entangled quantum state generation method based on the PPT criterion, which greatly improves the generation efficiency while ensuring the randomness of the generation of the quantum state density matrix. Background Art

[0002] Quantum entanglement is one of the most important resources in quantum information. It enables a stronger correlation between two or more subsystems in a system than in any classical way [1]. Because of this property, quantum entanglement plays a key role in many quantum information processing tasks, such as quantum teleportation and quantum key distribution [2,3].

[0003] The density matrix is ​​a very powerful tool in quantum mechanics. If the quantum state is a pure state, it can be described by a state vector. If the quantum state is a mixed state (which is also the state that needs to be touched in most cases in experiments), the density matrix needs to be used to describe it. The density matrix is ​​a basic object in quantum mechanics theory [4] and plays an extremely important role in quantum mechanics. It can not only describe the statistical mixed state of the system, but also effectively express the characteristics of the entangled state. The generation and operation of the density matrix is ​​an indispensable step in quantum information science. In particular, in practical quantum information processing tasks, how to generate a random quantum density matrix is ​​a fundamental task [5].

[0004] However, although the generation of density matrices can be achieved through a variety of random methods, determining whether the quantum states corresponding to these density matrices are entangled states is a complex problem. Even for a quantum state whose density matrix is ​​completely known, determining whether it is entangled has been proven to be an NP-hard problem [6]. Many scholars have proposed various criteria for entanglement determination, among which the most famous is the Positive Partial Transpose (PPT) criterion [7]. The PPT criterion states that if the partial transpose of a quantum state is not positive definite, then the state must be an entangled state. However, the PPT criterion is only complete for a two-qubit system. In higher-dimensional systems, it can only identify a portion of entangled states, and it cannot identify certain special entangled states (such as bound entangled states [8]).

[0005] In response to this problem, many researchers have proposed various improved entanglement determination criteria in recent years [9,10]. Although these criteria have improved the accuracy of entangled state determination to a certain extent, they still cannot cover all possible entangled state types, especially for entangled state determination in multi-body quantum systems, which is still full of challenges.

[0006] In recent years, the rise of artificial intelligence technology has provided a new way to solve problems in various fields. Similarly, research on quantum entanglement using machine learning has also continued to emerge [11,12]. Through machine learning, researchers can build entangled state classifiers to classify quantum states in an automated manner. Specifically, using machine learning models such as artificial neural networks, researchers can train a large number of labeled entangled state and separable state samples to obtain a model that can effectively determine quantum entanglement. For example, using a trained neural network classifier, researchers can quickly classify new random quantum state samples with a high accuracy, without relying on traditional judgment criteria

[13] .

[0007] For a machine learning solution, the three most important parts are: data features, data labels, and model structure. Labeling samples in a data set often consumes a lot of resources. For example, for the task of detecting true entanglement, it is necessary to generate a sufficient number of two types of random samples at the same time, one is truly entangled samples, and the other is separable samples. The generation of separable samples is relatively simple, but there is still no good unified method for generating entangled samples. Although there are many criteria for detecting true entanglement [14,15], it is still very time-consuming to generate a sufficient number of truly entangled samples in a completely random sample space.

[0008] Therefore, developing a method that can effectively improve the efficiency of generating random true entangled states is of great significance to the machine learning model that studies quantum entanglement. Although there are many criteria for entanglement determination (such as PPT criterion, entropy criterion, etc.), the time resources consumed by these methods to generate a large number of true entangled samples in a completely random quantum state sample space are unbearable. Therefore, finding a method that can efficiently generate entangled states in a wider range of random quantum state spaces can not only provide more training data for machine learning models, but also accelerate the progress of quantum entanglement research.

[0009] References

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[0024]

[15] Tóth,G.(2012).Multipartite entanglement and high-precisionmetrology.Physical Review A,85(2),022322. Summary of the invention

[0025] The present invention aims at the problem of low efficiency in the process of generating random true entangled states, and proposes a method for generating random true entangled quantum states based on the PPT (partial transposition positive definiteness) criterion. The core idea of ​​the method is to introduce a pre-screening step on a completely randomly generated density matrix, partially transpose each subsystem or partition of the quantum state, calculate the minimum eigenvalue of the transposed density matrix, and set a negative threshold value. Only when the minimum eigenvalue is less than the threshold value, the complex entanglement judgment criterion is further used to detect the true entanglement of the quantum state, thereby greatly improving the efficiency of generating true entangled states.

[0026] Traditional random quantum state generation methods usually directly input the randomly generated density matrix into the entanglement judgment criterion to detect whether the quantum state is a true entangled state one by one. This process requires the use of a relatively complex and time-consuming entanglement judgment algorithm in large-scale random samples, so the computational overhead is extremely large, especially when the scale of the quantum system is large, the computational complexity will increase exponentially. This direct generation method not only consumes a lot of computing resources, but also causes each true entangled state sample to take a very long time to generate. Therefore, solving this efficiency problem is of great significance for the large-scale generation of random entangled state samples.

[0027] The specific steps are as follows:

[0028] Step 1: Generate a random complex matrix and preprocess it to satisfy the properties of the density matrix to obtain a density matrix ρ;

[0029] The density matrix needs to satisfy the following properties: 1. Hermitian property: the density matrix is ​​equal to its own conjugate transpose; 2. Semi-positive definiteness: all eigenvalues ​​of the density matrix are non-negative; 3. Normalization: the trace of the density matrix is ​​equal to 1;

[0030] Step 2: Partially transpose the density matrix between each subsystem and between each partition, and then calculate their minimum eigenvalues ​​respectively;

[0031] Step 3: Set a threshold value, initially 0, and determine whether each minimum eigenvalue is less than the threshold value. If not, return to step 1 to regenerate a density matrix ρ. If yes, proceed to the next step.

[0032] Step 4: Select a true entanglement judgment criterion, perform true entanglement detection on the density matrix that has passed the pre-screening, and proceed to the next step if true entanglement is detected; if not, return to step 1;

[0033] Step 5: Record the time required to generate a true entangled quantum state under the threshold value, and then gradually reduce the threshold value, for example, by 0.005 each time;

[0034] Step 6: Use the lowered threshold value to repeat the above steps, gradually reduce the threshold value and the reduction amplitude, and repeat steps 1 to 5 until the threshold value with the shortest time to generate a truly entangled quantum state is found. This is the optimal pre-screening threshold value under this entanglement judgment criterion.

[0035] The advantages of the present invention are:

[0036] 1) The present invention is a random method for generating truly entangled quantum states based on the PPT criterion. By utilizing the easy calculation and good operating characteristics of the PPT criterion, the efficiency of generating truly entangled quantum states is greatly improved, and it has wide application value in various quantum information processing tasks.

[0037] 2) The present invention is a random true entangled quantum state generation method based on the PPT criterion, which can be effectively applied to various random quantum spaces and entanglement judgment criteria. As the quantum space increases, its generation efficiency is more significantly improved, and it has strong scalability. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 This is a flow chart of the random truly entangled quantum state generation method of the present invention.

[0039] Figure 2 A comparison diagram of the time consumed and the threshold value for preparing a true entangled quantum state in a three-qubit system for the present invention. DETAILED DESCRIPTION

[0040] The technical solution of the present invention is further described below in conjunction with embodiments.

[0041] The present invention proposes a random true entangled quantum state generation method based on the PPT criterion. The core idea of ​​this method is to use the scalability of the PPT criterion in three-body or even multi-body quantum states, and perform partial transposition between subsystems and partitions on the quantum states in the random quantum space for pre-screening. This greatly reduces the number of times the more time-consuming entanglement criterion is applied, reduces the average generation time of each true entangled quantum state, and improves the overall generation efficiency.

[0042] Figure 1 A flow chart of the present invention is given, and the specific steps are as follows:

[0043] Step 1: First, randomly generate a complex matrix and preprocess it to satisfy the basic properties of the density matrix, and finally obtain a density matrix ρ that meets the description of the quantum state;

[0044] The density matrix needs to satisfy the following three important properties:

[0045] 1. Hermitian property: The density matrix is ​​equal to its own conjugate transpose, that is:

[0046] ρ=ρ + (1)

[0047] 2. Semi-positive definiteness: All eigenvalues ​​of the density matrix are non-negative, that is, for any state vector |ψ>, we have:

[0048] <ψ|ρ|ψ>≥0 (2)

[0049] 3. Normalization: The trace of the density matrix is ​​equal to 1, that is:

[0050] Tr(ρ)=1 (3)

[0051] It is worth mentioning that there are two different ways to generate random density matrices:

[0052] 1. Randomly generate matrix elements: Independently randomly generate the real and imaginary parts of each element of the density matrix. These random numbers can be generated using Gaussian distribution or uniform distribution. Pay attention to the random generation range of the real and imaginary parts, because the randomness of all elements will affect the size of the random sample space. If certain parameters are limited, such as making the real and imaginary parts positive, the range of the sample space will be significantly reduced.

[0053] 2. Generate quantum state based on unitary matrix: First, randomly generate a unitary matrix U, and then use the unitary matrix to act on the initial state to obtain a random pure state |ψ>, whose density matrix is:

[0054] ρ=|ψ><ψ| (4)

[0055] Then, by mixing i such random pure states, the density matrix of the target mixed state can be obtained:

[0056]

[0057] Among them, Σ i p i =1.

[0058] Step 2: Partially transpose the density matrix between each subsystem and between each partition, and then calculate their minimum eigenvalues ​​respectively;

[0059] For example, for a three-qubit system A, B, and C, each subsystem is partially transposed. Assuming that system A is transposed, its partial transposition Defined as:

[0060]

[0061] That is, the matrix elements of system A are swapped, while the other systems (B and C) remain unchanged. Similarly, the B and C systems can be transposed.

[0062] For a three-qubit system, there are three possible partial transpositions: But if our goal is a four-qubit quantum state with four subsystems A, B, C, and D, there are seven possible cases: (1) transposing individual subsystems A, B, C, and D, for a total of four cases; (2) transposing two subsystems simultaneously (such as AB, AC, and AD), for a total of three cases.

[0063] For each partial transposition, calculate its minimum eigenvalue. The specific calculation process is: solve the eigenvalue of the partial transposition density matrix and calculate the minimum eigenvalue of each part. If the minimum eigenvalue is less than zero, it indicates that the quantum state may have entanglement characteristics.

[0064] Step 3: Set an initial threshold value η, for example η = 0. For each partial transposition result, determine whether its minimum eigenvalue is less than the threshold value: (1) If the minimum eigenvalue of the density matrix of all partial transpositions is less than η, it is considered that the state has passed the pre-screening and proceeds to the next step; (2) If any eigenvalue is greater than or equal to η, regenerate the density matrix and repeat steps 1 and 2.

[0065] Step 4: For quantum states that pass the pre-screening, use a more stringent entanglement detection criterion to perform true entanglement detection on them. If the state is detected to be a true entangled state, proceed to the next step; if no true entanglement is detected, return to step 1.

[0066] Step 5: Record the time required to generate a true entangled quantum state under the current threshold value. To optimize the generation efficiency, after each detection, reduce the threshold value η, for example, by 0.0050 each time, and re-execute the above steps.

[0067] Step 6: Test the efficiency of generating true entangled states at different threshold values ​​by gradually reducing the threshold value η and the step size, until the threshold value η with the shortest time required to generate true entangled states is found. opt This optimal threshold value will serve as the most effective criterion in the pre-screening process and will be used for the subsequent generation of truly entangled quantum states.

[0068] Through the method of the present invention, truly entangled quantum states can be effectively screened out while reducing the amount of calculation, thereby greatly improving the generation efficiency.

[0069] By way of example, the specific implementation process of the method of the present invention is described below with specific examples.

[0070] This embodiment uses a three-qubit system as an example to illustrate the implementation process: First, a complex matrix is ​​randomly generated, and the real and imaginary parts of the complex numbers can come from a Gaussian distribution or a uniform distribution. For a three-qubit system, its density matrix is ​​an 8×8 matrix. The generated complex matrix is ​​preprocessed so that it satisfies the basic properties of the density matrix: Hermitian, semi-positive definiteness, and normalization. Each subsystem is then partially transposed. The definition of partial transposition is to transpose the matrix elements of a subsystem while keeping other subsystems unchanged. Here, we take systems A, B, and C as examples for partial transposition. When partially transposing system A, we transpose the matrix blocks related to A, keeping other subsystems B and C unchanged. After completing the partial transposition, the minimum eigenvalue of the transposed density matrix is ​​calculated. And compared with the set initial threshold value, if the minimum eigenvalues ​​of all partial transpositions are less than the threshold value, it indicates that the density matrix has passed the pre-screening and enters the next step of entanglement detection. For density matrices that pass the pre-screening, a more complex true entanglement judgment criterion is used to perform true entanglement detection on them. If the density matrix is ​​judged to be a true entangled state, the time it takes to generate a true entangled state under the current threshold value is recorded. Then the threshold value is reduced by a fixed amount (for example, 0.005), and the reduced threshold value is continued to be used, and the above steps are repeated. Through multiple iterations, the threshold value with the shortest time required to generate a true entangled state is found. At this time, the threshold value is the optimal pre-screening threshold value. Figure 2 This is the implementation result diagram of the three-qubit system.

[0071] Figure 2 It shows that, through the method of the present invention, a threshold value can be found to increase the probability of screening true entangled quantum states. Specifically, compared with not setting any threshold (i.e., the threshold value is 0), the method of the present invention has a significant improvement. When the threshold value η = -0.012, the efficiency of generating true entangled quantum states is increased by 71.67% compared with not setting the threshold value η = 0.

[0072] Further analysis shows that when the threshold value is 0, all generated density matrices will be calculated and processed, and the lack of any screening strategy will result in a large number of non-true entangled states being involved in the calculation, wasting a lot of computing resources. After introducing an appropriate threshold value, some density matrices that do not meet the true entanglement conditions can be eliminated at an early stage, which significantly reduces redundant calculations in subsequent steps. The introduction of the threshold value optimizes the entire process, thereby significantly improving the efficiency of generating true entangled states while ensuring the accuracy of the results.

[0073] Therefore, by reasonably setting the threshold value, the time required to generate a true entangled state in a three-qubit system is significantly optimized, demonstrating the effect of the present invention in quantum state generation.

Claims

1. A method for generating random true entangled quantum states based on the PPT criterion, the specific steps comprising: First, a complex matrix is ​​randomly generated and preprocessed to satisfy the basic properties of the density matrix, thereby obtaining a density matrix ρ that meets the description of the quantum state, and using this density matrix to represent the generated quantum state; Secondly, the density matrix ρ is partially transposed between each subsystem and between each partition, and its minimum eigenvalue is calculated respectively; By judging the relationship between the minimum eigenvalue and the threshold value, the randomly generated quantum state is pre-screened; specifically, if the minimum eigenvalue of all partially transposed density matrices is less than the threshold value, it is considered to have passed the pre-screening and proceed to the next step; otherwise, the density matrix is ​​regenerated and pre-screened; Then, for the quantum states that pass the pre-screening, a more stringent entanglement detection criterion is used to detect true entanglement; if the detection is a true entangled state, proceed to the next step; if no true entanglement is detected, the density matrix is ​​regenerated and the above steps are repeated; Next, record the time required to generate a truly entangled quantum state under the current threshold value; After each detection, the threshold value is gradually reduced and the above steps are performed again; Finally, the efficiency of generating true entangled states under different threshold values ​​is tested until the threshold value with the shortest time required to generate true entangled states is found; This optimal threshold value will serve as the most effective criterion in the pre-screening process for the subsequent generation of truly entangled quantum states.

2. A method for generating random true entangled quantum states based on the PPT criterion as claimed in claim 1, characterized in that: The density matrix ρ satisfies (1) Hermitian property: The density matrix is ​​equal to its own conjugate transpose, that is: ρ = ρ + ; (2) Semi-positive definiteness: All eigenvalues ​​of the density matrix are non-negative, that is, for any state vector ψ>, we have: ψ|ρψ>≥0; (3) Normalization: The trace of the density matrix is ​​equal to 1, that is, Tr(ρ) = 1.

3. A method for generating random true entangled quantum states based on the PPT criterion as claimed in claim 1, characterized in that: The threshold is initially set to 0, and then after the true entanglement detection of the generated quantum state is performed, the threshold is gradually reduced until the threshold value with the shortest time required to generate the true entangled state is found.

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