Turbulent atmosphere target detection method based on dual-mode compressed vacuum state quantum illumination radar

Abstract

The invention belongs to the technical field of quantum radar detection, and relates to a turbulent atmosphere target detection method based on a dual-mode compressed vacuum state quantum illumination radar, and the method comprises the steps: selecting a TMSV state as an emission light field according to the working principle of the quantum illumination radar, and describing the TMSV state as a TMSV state density matrix; establishing a model for revealing a physical decoherence mechanism according to an influence mechanism of an atmospheric turbulence effect on signal mode light field propagation; in the propagation process of the signal mode light field in the turbulent atmosphere, a lost quantum state density matrix is obtained by acting a Kraus operator on the TMSV state density matrix; a density matrix of a diffused quantum state is obtained by acting a Kraus operator on the density matrix of the TMSV state; modeling a target as a beam splitter, and describing the interaction of a signal mode light field and background thermal noise at the target by using a unitary operator to detect the target; the detection error probability is reduced, and the detection sensitivity is improved.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of quantum radar detection, and particularly relates to a turbulent atmosphere target detection method based on a two-mode squeezed vacuum state quantum illumination radar. BACKGROUND

[0002] The development of radar technology can be traced back to the exploration of electromagnetic wave properties in the late 19th century. By emitting high-frequency electromagnetic pulses and measuring the round-trip time of the target echo, radar can accurately determine the distance of the target; if combined with the pointing information of the antenna, the azimuth angle of the target can be further determined to realize the positioning of the target. As an important electromagnetic sensor, modern radar has been widely used in long-distance detection, positioning, tracking and identification of various target objects.

[0003] Although the traditional radar technology is quite mature and plays a key role in military and civilian fields, its detection performance and anti-interference ability in complex environments still face many challenges. On the one hand, in the complex battlefield environment with strong clutter and noise, it is difficult for traditional radar to effectively distinguish target signals from background interference. Especially under the condition of low transmission power or strong background noise, the target echo signal is easily overwhelmed by noise, resulting in a sharp decline in the detection sensitivity and overall performance of the radar. On the other hand, the detection ability of traditional radar on stealth targets is limited. Modern stealth technology reduces the radar scattering cross-section of the target, making the reflection of radar waves by stealth targets such as stealth aircraft significantly weakened, which makes it difficult for traditional radar to effectively detect such low-detectability targets. In addition, in the electronic countermeasure environment, traditional radar is also vulnerable to interference and deception by the enemy. The enemy can confuse the radar by emitting strong interference signals or faking target signals (decoys), making it difficult for the radar to distinguish real target echoes.

[0004] In order to overcome the limitations of traditional radar, researchers have proposed the concept of quantum radar in recent years, aiming to improve the detection performance of radar in complex environments using quantum mechanics. Quantum radar is considered a new long-range sensing technology based on quantum effects such as quantum coherence and quantum entanglement, which is expected to break through the performance limits of classical radar. According to the working principle, quantum radar can be mainly divided into three categories: quantum interference radar, quantum enhancement radar, and quantum illumination radar. Among them, the quantum illumination radar scheme proposed by Lloyd in 2008 has become a research hotspot due to its excellent performance in noisy environments. Literature (Karsa A, Fletcher A, Spedalieri G, et al. Quantum illumination and quantum radar: a brief overview [J]. Reports on Progress in Physics, 2024, 87(9): 094001) shows that the signal-to-noise ratio of quantum illumination radar is significantly improved compared to the classical benchmark, and the error probability is 6 dB lower than the classical benchmark.

[0005] In recent years, researchers have actively explored and evaluated a variety of quantum states in order to find the ideal light field that can maximize the performance of quantum illumination radar. Early research focused on tapping the potential of various non-classical light fields, such as single photons and entangled photon pairs, hoping to surpass the performance of classical coherent state light fields. Shapiro and Lloyd's groundbreaking theoretical work compared the performance differences between single-photon quantum illumination radar and traditional coherent state detection schemes. Devi and Rajagopal simplified the calculation method of error probability in quantum channel recognition, systematically comparing the detection performance of different quantum states, including Fock states, NOON states, and coherent states. These early studies not only laid the theoretical foundation for quantum illumination, but also initially revealed the potential advantages of quantum methods in target detection, pointing the way for further exploration of more complex and potentially powerful quantum states.

[0006] Among the numerous candidate quantum states, the two-mode squeezed vacuum state (TMSV) is favored for its good robustness in noisy and lossy environments. Even in the presence of severe environmental noise and significant channel loss, the intrinsic quantum correlations of TMSV can still bring performance improvement for target detection. For example, Fan and Zubairy studied the difference in target detection performance between the bright and dim TMSV states in quantum illumination radar; Tao and Ren compared the performance of three types of entangled coherent states and TMSV states in quantum illumination radar; and Allahverdi and Motazedifard

[11] analyzed the theoretical detection range of TMSV state quantum radar in detail and proposed the potential application of TMSV state quantum radar in long-range detection. These research results generally show that, compared with other quantum states, TMSV state exhibits more excellent comprehensive performance in target detection.

[0007] However, the interference of the atmospheric environment with the transmission of quantum optical fields is one of the key bottlenecks restricting the practical application of quantum radar. In actual target detection tasks in the atmosphere, the optical quantum signals emitted by the quantum radar will inevitably interact with the dust, particles, aerosols and other components widely distributed in the atmosphere, undergo scattering and absorption, and thus cause photon loss. At the same time, the random distribution of atmospheric temperature and wind speed will form turbulent vortices, causing random fluctuations in the refractive index of the atmosphere. This causes the transmitted optical field to experience atmospheric scintillation and phase jitter and other turbulence effects. These effects further cause the entanglement of the quantum optical field to decrease during transmission, causing it to gradually lose its original quantum characteristics and degenerate into a classical optical field, ultimately significantly reducing the key performance indicators of the quantum radar system, such as detection sensitivity and resolution.

[0008] Currently, classical radar can achieve high-precision optical synthetic aperture imaging in the kilometer range through active optical intensity interference technology and robust image restoration algorithms. However, for quantum radar, although the academic community has carried out a lot of research on its scheme design and theoretical performance, the in-depth exploration of the influence mechanism of atmospheric turbulence on quantum radar performance is still insufficient, and the existing research focuses more on the simplified analysis of atmospheric loss. For example, Li et al. studied the detection performance of the partially post-selected filtering quantum radar, and introduced Gaussian white noise to simulate the influence of the atmosphere. Shi et al. systematically studied the performance of quantum radar under actual atmospheric conditions, analyzed the influence of atmospheric loss, and pointed out that increasing the signal bandwidth can help achieve a longer detection distance. D. Vasylyev et al. proposed a model based on elliptical light field approximation to describe the influence of atmospheric turbulence on quantum channels, and derived the transmittance probability distribution function under weak, weak to moderate and strong turbulence conditions, and verified its physical consistency and applicability compared with the lognormal model through experiments. Although the existing research shows that quantum radar has significant detection advantages, its practical application still needs to overcome a series of key technical challenges such as efficient generation and stable maintenance of entangled states, loss and noise interference in the signal transmission process, and adaptability of the system to complex environments. Therefore, future research needs to further explore the performance of quantum radar in real application scenarios and actively seek effective ways to improve its performance, in order to accelerate the practicalization process of this technology. SUMMARY

[0009] The purpose of the present application is to overcome the deficiencies in the prior art and provide a turbulent atmospheric target detection method based on a two-mode compressed vacuum state quantum illumination radar.

[0010] To achieve the purpose of the present application, the technical solutions adopted by the present application are as follows.

[0011] A turbulent atmospheric target detection method based on a two-mode compressed vacuum state quantum illumination radar, comprising the following steps:

[0012] S1. According to the working principle of the quantum illumination radar, a TMSV state is selected as the transmission light field of the quantum illumination radar, and the transmission light field is described as a TMSV state density matrix ρ AB ;

[0013] S2. According to the influence mechanism of atmospheric turbulence on the propagation of the signal mode light field, a model revealing the physical dephasing mechanism is established; wherein: the model is linearly superimposed by an atmospheric dissipation part described by an atmospheric dissipation rate and a phase diffusion part described by a phase diffusion rate;

[0014] S3. During the propagation of the signal mode light field in the turbulent atmosphere, the atmospheric dissipation part and the relationship between the atmospheric dissipation rate and the transmittance of the turbulent atmosphere are used to obtain the TMSV state density matrix ρ ABacting Kraus operators, obtaining quantum state density matrix after loss Using the phase diffusion part and the dimensionless parameter estimated by Fried parameter and turbulence outer scale wave number, by means of TMSV state density matrix ρ AB acting Kraus operators, obtaining quantum state density matrix after diffusion

[0015] S4, modeling the target as a beam splitter with transmissivity η t and using the unitary operator U BS Describing the interaction of signal mode light field and background thermal noise at the target, that is, the mixing process;

[0016] S5, assuming that the to-be-detected region has two cases: target existence and target nonexistence:

[0017] If the target exists, the signal mode light field and the background thermal noise interact at the target, and after being reflected to the receiving end, a joint state ρ1 is generated with the reference mode light field;

[0018] If the target does not exist, only the background thermal noise is reflected to the receiving end, and a joint state ρ0 is generated with the reference mode light field.

[0019] Further, the TMSV state density matrix ρ AB is obtained by a wave function of the TMSV state, and the expression is: ρ AB = |ψ TMSV ><ψ TMSV |, |ψ TMSV > is the right vector of the TMSV state.

[0020] Further, the TMSV state is generated by a two-mode squeezing operator acting on a vacuum state, and the expression is: In the formula, |00> AB represents a vacuum state, and S2(r) represents a two-mode squeezing operator:

[0021] Further, the influence mechanism includes amplitude fluctuation and phase fluctuation.

[0022] Further, the model is:

[0023]

[0024] In the formula: is an annihilation operator, is a photon number operator, γ A is a dissipation rate, and Γ A is a phase diffusion rate.

[0025] Further, the quantum state The expression of U is:

[0026]

[0027] In the formula: are the annihilation operators corresponding to signal mode A and idler mode B, respectively.

[0028] Further, the quantum state after loss The derivation process is as follows:

[0029] Considering the amplitude fluctuation and the loss effect caused thereby. The random inhomogeneity of refractive index caused by atmospheric turbulence will cause distortion of the wavefront of the transmitted light beam, resulting in a random light intensity distribution on the receiving plane, i.e. atmospheric scintillation effect. This effect makes the transmittance of the atmospheric channel no longer a fixed value, but a random variable. For the transmitted quantum state, this is equivalent to experiencing a random loss channel. Considering a two-mode light field ρ AB The loss process during transmission in the atmosphere can be described by the two-mode form of the amplitude fluctuation part in model (5):

[0030]

[0031] In the formula: parameters γ A and γ B and the atmospheric transmittance T have the following relationship:

[0032]

[0033] For the atmospheric transmittance T A of signal mode A, and the atmospheric transmittance T B of reference mode B, when stored locally, T B = 1, the quantum state after loss can be obtained by acting the corresponding Kraus operator on the initial state ρ AB :

[0034]

[0035] After that, we can get:

[0036]

[0037] Further, the expression of the unitary operator U BS is:

[0038]

[0039] In the formula: and are the annihilation operators of the signal mode light field and the thermal noise mode thermal field, respectively.

[0040] Further, the expression of the joint state ρ1 is:

[0041]

[0042] In the formula, Tr C Perform partial trace operation on system C to extract subsystem information from the composite system density matrix; U BS Describe the unitary operator of the beam splitter transformation, which corresponds to the mixing and reflection of the optical field at the target output; Tensor product symbol for constructing the density matrix of the composite system, describing the quantum state of the composite system composed of multiple subsystems; ρ' AB The density matrix of the two-mode entangled state after passing through the turbulent flow.

[0043] Further, the expression of the joint state ρ0 is:

[0044]

[0045] Tr A Perform partial trace operation on system A; Tensor product symbol for constructing the density matrix of the composite system, describing the quantum state of the composite system composed of multiple subsystems; ρ' AB The density matrix of the two-mode entangled state after passing through the turbulent flow.

[0046] Further, the η t Represents the effective reflectivity of the target (0 ≤ η t ≤ 1)

[0047] Compared with the prior art, the beneficial effects of the present application are:

[0048] Improve the target detection performance in complex environment: in complex environments such as turbulent atmosphere, the detection of weak targets shows excellent comprehensive performance, which can effectively resist the influence of atmospheric attenuation and phase diffusion, reduce the detection error probability, improve the detection sensitivity, and break through the performance bottleneck of classical radar in strong clutter and low power conditions;

[0049] A noise model containing atmospheric attenuation and phase diffusion is constructed, and the influence of its on the detection performance of quantum radar is analyzed, which reveals the influence law of entanglement reduction caused by turbulence on the detection error probability, and the significant influence of phase diffusion channel on system performance, including the non-monotonic behavior of performance saturation or even reversal in the high average number of transmitted photons region, which provides important theoretical guidance for the design, deployment and parameter optimization of quantum radar in real atmospheric environment. BRIEF DESCRIPTION OF DRAWINGS

[0050] Figure 1This is a flow chart of a method for batch program downloading of an ESC system controller according to the present invention;

[0051] Figure 2 is the average number of photons in the TMSV state after the dissipation channel under different atmospheric parameters <N out >Surface plot, describing the average transmittance and flicker index Impact on photon number loss, initial emission photon number N emit =1, the cutoff dimension of numerical calculation is 10;

[0052] Figure 3 is the von Neumann entropy E under the atmospheric dissipation channel with the average number of emitted photons N emit The atmospheric parameters are set to: average transmittance Flicker Index Beam splitter transmittance η t =0.01, the average number of photons in the thermal light field N th =0.1, the cutoff dimension used in numerical calculations is 10;

[0053] Figure 4 is the target detection performance under atmospheric dissipation channel with the average number of emitted photons N emit where: (a) Helstrom limit error probability P err,M ; (b) Quantum Chernoff index ξ QCB , the atmospheric parameters are set as: average transmittance Flicker Index Beam splitter transmittance η t =0.01, the average number of photons in the thermal light field N th =0.1, the truncation dimension used in numerical calculations is 10;

[0054] Figure 5 is the diffusion intensity of different phases Γ a The von Neumann entropy decreases with the average number of emitted photons N emit The change diagram of the beam splitter transmittance η t =0.01, the average number of photons in the thermal light field N th =0.1, the cutoff dimension used in numerical calculations is 10;

[0055] Figure 6 is the diffusion intensity of different phases Γ a The target detection performance under the condition of average number of emitted photons N emit where: (a) Helstrom limit error probability P err,M ; (b) Quantum Chernoff index ξ QCB . The number of entangled light fields M = 1, the beam splitter transmittance η t=0.01, the average number of photons in the thermal light field N th =0.1, the cutoff dimension used in numerical calculations is 10;

[0056] Figure 7 is the calculated result of the Helstrom limit error probability under the influence of phase diffusion channel under high photon number; where: (a) Helstrom limit error probability P err,M As the average number of emitted photons N emit The curve of the function shows the diffusion intensity Γ at different phases a The high photon number behavior under the condition of Γ a = 0.05, 0.1, 0.15, 0.2 and 0.5. The ★ mark is the minimum point of the curve. The number of entangled light fields M = 1, the beam splitter transmittance η t =0.01, the average number of photons in the thermal light field N th = 0.1, the cutoff dimension used in the numerical calculation is 40; (b) The optimal average number of emitted photons N opt The curve that converges with the increase of truncation dimension shows the optimal average number of emitted photons N when the truncation dimension continues to increase. opt Gradually converge to a stable value;

[0057] Figure 8 The Helstrom limit error probability minimum point characteristic changes with the phase diffusion intensity Γ a The left vertical axis (square dot mark) represents the optimal average number of emitted photons N required to achieve the minimum error probability. opt The right vertical axis (triangle mark) indicates the corresponding minimum Helstrom limit error probability The number of entangled light fields M = 1, the beam splitter transmittance η t =0.01, the average number of photons in the thermal light field N th =0.1, and the cutoff dimension used in the numerical calculation is 40. DETAILED DESCRIPTION

[0058] 1. Basic Principles of Quantum Illumination Radar

[0059] The core task of quantum illumination radar is to detect low reflectivity targets under strong background noise interference. Its basic working principle is as follows Figure 1 As shown in the figure, the key lies in leveraging the properties of quantum entanglement to enhance detection sensitivity. The system generates a dual-mode entangled light field at the transmitter end, where one beam of light (called signal mode A) is sent to the area to be detected, while the other beam of light (called reference mode B, or idle mode) is stored locally.

[0060] The detection procedure usually involves two hypothetical scenarios. When the target is absent in the detection region, the signal mode A is completely lost during the transmission process and cannot return to the receiving end, in which case the receiving end can only receive the environmental background noise. When the target exists in the detection region, the signal mode A is partially reflected after being irradiated onto the target, and the reflected echo is mixed with the background thermal noise of the target region before returning to the receiving end.

[0061] At the receiving end, the returned signal light (if any) will be jointly quantum measured with the locally stored reference mode B. Since the signal mode A and the reference mode B are initially in an entangled state, the signal mode A experiences loss and noise pollution during the transmission and reflection processes, and some quantum correlation between the two may still remain. By designing and implementing a precise joint measurement strategy to distinguish the different quantum states received under the two hypothetical scenarios, a decision can be made as to whether the target exists or not.

[0062] For ease of theoretical analysis, the target is usually modeled as a beam splitter with a transmittance of η t , where η t represents the effective reflectivity of the target (0≤η t ≤1). The interaction (i.e., the mixing process) between the signal mode A and the background thermal noise (represented by mode C) at the target can be described by the unitary operator U BS of the beam splitter:

[0063]

[0064] Based on the above model, before the joint measurement is performed, the quantum state of the system obtained by the receiving end corresponds to two hypotheses:

[0065] Target absent (H0):

[0066]

[0067] Target present (H1):

[0068]

[0069] where ρ AB is the density matrix of the initial two-mode entangled state generated by the transmitting end, and ρ C is the density matrix of the background thermal noise (usually assumed to be in a thermal state), which is represented as follows:

[0070]

[0071] where N th is the average photon number of the thermal light field. By performing an optimal or near-optimal quantum state discrimination measurement on the two quantum states ρ0 and ρ1, target detection can be achieved.

[0072] II. Transmission model in turbulent atmosphere

[0073] When quantum radar signals are transmitted in the real atmosphere, they are inevitably affected by atmospheric turbulence, which leads to signal attenuation and entanglement reduction, and further reduces the detection performance of the radar. Atmospheric turbulence mainly affects the propagation of light fields through two mechanisms: one is to cause amplitude fluctuations, leading to additional transmission loss; the other is to cause phase fluctuations, leading to phase diffusion. When considering the effects of random loss and phase fluctuations in single-mode cases, it can be described as the following master equation:

[0074]

[0075] First, we consider the amplitude fluctuations and the loss effect caused by them. The random inhomogeneity of the refractive index caused by atmospheric turbulence will cause the wavefront of the transmitted light field to be distorted, resulting in a random light intensity distribution on the receiving plane, i.e., the atmospheric scintillation effect. This effect makes the transmittance of the atmospheric channel no longer a fixed value, but a random variable. For the transmitted quantum state, this is equivalent to experiencing a random loss channel. Consider a two-mode light field ρ AB In the atmosphere, the loss process can be described by the two-mode form of the amplitude fluctuation part in the master equation (5):

[0076]

[0077] The parameters γ A and γ B in the formula have the following relationship with the atmospheric transmittance T:

[0078]

[0079] For the atmospheric transmittance T A (of the signal mode A) and T B (of the reference mode B, saved locally T B = 1), the quantum state ρ(T A , T B ) after experiencing loss can be obtained by acting on the initial state ρ AB the corresponding Kraus operator:

[0080]

[0081] Then we get:

[0082]

[0083] The key to the problem of atmospheric dissipation is that the transmittance T A of the signal mode A is affected by atmospheric turbulence.becomes a random fluctuation. To describe this randomness, a probability density function of transmission coefficient (PDTC) is needed. Based on the Kolmogorov turbulence theory, the PDTC can be approximated by a lognormal distribution model under weak turbulence conditions:

[0084]

[0085] where the mean transmission coefficient and the scintillation index are given by

[0086]

[0087] Here ω0is the waist radius of the transmitted optical field, D is the diameter of the receiving aperture, λ is the wavelength of light, L is the propagation distance, is the refractive index structure constant (characterizing the turbulence strength), and k = 2π / λ is the wave number. mainly reflects the influence of the diffraction loss and the size of the receiving aperture on the mean transmission coefficient, while quantifies the fluctuation strength of the transmission coefficient caused by turbulence. When evaluating the average performance of the system, the calculation results of the performance indicators depending on T A need to be integrated and averaged under the P(T A ) distribution.

[0088] Secondly, the atmospheric turbulence also causes random fluctuations in the phase of the optical field. When the size of the optical field is much smaller than the turbulence coherence length r0(i.e., ω0<< r0), the entire optical field can be approximated as experiencing a uniformly random phase drift. This effect is called phase diffusion, and its influence on the quantum state evolution can be described by the two-mode form of the corresponding part in the master equation (5):

[0089]

[0090] where is the photon number operator, and Γ A , Γ B is the phase diffusion rate. The cumulative phase diffusion effect after a time t can be characterized by a dimensionless parameter Γ a = Γ A t. This parameter is closely related to the atmospheric conditions and can be estimated by the Fried parameter r0and the turbulence outer scale wave number κ0:

[0091] Γ a = 1.09(r0κ0) -5 / 6 (14)

[0092] The specific influence of the phase diffusion process on the evolution of the quantum state can also be calculated by the corresponding Kraus operators acting on the density matrix:

[0093]

[0094] In the following, we study the influence of random loss and phase diffusion on the system performance, respectively. The combined effect of these two factors, which leads to the degradation of quantum entanglement, is the key physical factor affecting the detection performance of quantum radar in real atmospheric environments.

[0095] 2.3 Two-mode squeezed vacuum state

[0096] Quantum illumination radar requires entangled light fields during operation. The selection of the entangled state of the entangled light field affects its performance to some extent. In the theory and application of quantum illumination radar, it is often necessary to express complex entangled states in a more general form for the analysis and comparison of different quantum states. In this case, the quantum state can be described by the superposition of Fock states (also known as photon number states). We usually want to write the quantum state in the following general form:

[0097]

[0098] In the formula, |n> A and |m> A represent the Fock states containing n and m photons in two modes (A and B), respectively. The coefficient C nm is the probability amplitude of the state with photon numbers n and m in the two modes.

[0099] In order to effectively resist the interference of channel loss and background noise, and fully utilize quantum correlation to enhance the detection ability, the present invention selects TMSV state as the transmission light field of quantum illumination radar. TMSV state is a typical entangled state in the field of continuous variable quantum optics, which can be generated by applying a two-mode squeezing operator to the vacuum state |00> AB :

[0100]

[0101] In the formula, r≥0 is the squeezing factor, which represents the squeezing degree and entanglement intensity; |n> A and |n> B are the Fock states (photon number states) of mode A and mode B, respectively.

[0102] ​The TMSV state is suitable for quantum illumination tasks mainly due to several key properties. First, it has strong quantum correlations between modes A and B, which are reflected in the highly nonclassical correlations between the measurement results of the photon number difference, the quadrature components, and the difference. Second, the TMSV state has good robustness against loss and noise. Even if one of the modes (e.g., the signal mode A that carries the detection task) experiences severe channel loss and noise pollution, the residual quantum correlations can still provide performance advantages over classical detection methods in the joint measurement stage. Finally, theoretical studies have shown that quantum illumination schemes based on TMSV states can achieve significantly improved target detection sensitivity, especially under low signal-to-noise ratio and high loss conditions, compared to traditional radars using classical coherent light fields.

[0103] In view of the advantages of the TMSV state, the present application constructs a theoretical model of a quantum illumination radar based on this quantum state and focuses on studying the target detection performance of the radar after considering the effects of turbulent atmospheric transmission.

[0104] 2.4 Quantum radar target detection performance indicators

[0105] In order to quantitatively evaluate the target detection performance of a quantum illumination radar based on TMSV states in a turbulent atmospheric environment, the present application selects the following three key performance indicators.

[0106] First, von Neumann entropy is selected. For a TMSV state, the von Neumann entropy of the reduced density matrix of the signal mode ρ A = Tr B |ψ AB ><ψ AB | can be used as a measure of entanglement:

[0107] E(ρ A ) = -Tr[ρ A lnρ A ] (18)

[0108] In the application scenario of quantum radars, by calculating the von Neumann entropy of the signal mode after transmission through the atmospheric channel, the degree of residual entanglement between the signal mode and the reference mode can be understood.

[0109] Second, to evaluate the theoretical limit of the radar's ability to distinguish between the two hypotheses of the existence (H1) and non-existence (H0) of a target, we introduce the Helstrom limit. It gives the theoretical lower bound of the minimum average error probability when distinguishing between two quantum states ρ0 and ρ1 with equal prior probabilities in the framework of quantum mechanics. Assuming that M pairs of entangled light fields are used for M independent detections, the total quantum states are and then the Helstrom limit error probability Perr,M For:

[0110]

[0111] where: denotes the trace norm of quantum state γ.

[0112] The Helstrom limit is the fundamental benchmark to measure the distinguishability of quantum radar systems, which takes into account both the miss-detection and false-alarm error types, and represents the best performance that can be theoretically achieved under the optimal quantum measurement strategy. The error probability of any practical measurement scheme must be greater than or equal to this limit value.

[0113] However, it is usually computationally very complex or even infeasible to directly calculate the Helstrom limit in the case of multiple copies (M > 1). Therefore, we introduce a more computationally tractable and practical upper bound on the error probability—quantum chernoff bound (QCB). The QCB provides an exponentially decaying upper bound on the error probability, which is usually a good approximation of the Helstrom limit, especially when the number of copies M is large. The form of the QCB error probability upper bound is as follows:

[0114]

[0115] where the quantum chernoff exponent

[0116]

[0117] describes the rate at which the error probability exponentially decreases with the number of probes M. The QCB exponent not only provides a practical index to measure the distinguishability of quantum states, but also facilitates numerical calculations and comparisons of the detection performance under different parameter conditions.

[0118] By calculating and analyzing the above three indicators, we can comprehensively evaluate the impact of turbulent atmospheric channels on the detection performance of quantum illumination radar based on TMSV states.

[0119] 3 Numerical simulation and results analysis

[0120] The target detection performance of TMSV state-based quantum illumination radar in turbulent atmospheric environment will be numerically simulated, and the effects of atmospheric dissipation channels and phase diffusion channels will be investigated. We focus on the changes in von Neumann entropy, Helstrom error limit, and QCB exponent under different average photon numbers (corresponding to changes in radar transmit power), and compare these results with the ideal case without considering the effects of atmospheric turbulence, in order to intuitively reveal the impact of atmospheric turbulence on radar target detection performance.

[0121] 3.1 Impact of atmospheric dissipation channels

[0122] In order to intuitively show the effect of atmospheric turbulence dissipation on the intensity of quantum light emitted from the emission end, we calculated the average transmittance at different and flicker index In this case, the expected number of photons in the TMSV state after atmospheric turbulence dissipation is as follows: Figure 2 As shown, the number of emitted photons N in the TMSV state emit = 1, at different average transmittances and flicker index Different intensity losses were generated under these conditions, with the average transmittance It is the main factor affecting the loss. When the average transmittance is the same, the loss of photons is small. Flicker Index The larger it is, the greater the loss of photons.

[0123] Figure 3 The curve of the von Neumann entropy of the TMSV state quantum illumination radar after it passes through the atmospheric dissipation channel and changes with the average number of photons is given. Figure 3 It can be clearly observed that the von Neumann entropy of the quantum state decreases due to the influence of the turbulent atmospheric dissipation channel, and the magnitude of the decrease increases with the average number of photons. This directly indicates that the presence of atmospheric turbulence causes the entanglement of the quantum state to decrease, thereby weakening the quantum correlations on which the radar system relies, ultimately leading to a decrease in detection performance. Figure 4 (a) shows how the Helstrom error limit changes with the average number of photons under the influence of the atmospheric dissipation channel. It can be seen that after passing through the atmospheric dissipation channel, the Helstrom limit error probability continues to decrease with the increase of the average number of photons, and the difference with the situation without considering the atmospheric dissipation channel is increasing, which means that the cost required to achieve the same detection error rate (such as transmission power) has increased. Figure 4 (b) shows the variation of the QCB index under the corresponding conditions. Compared to the ideal case, the QCB index decreases under the atmospheric dissipation channel (note that the QCB index is in negative logarithmic form; a decrease means an increase in the upper bound of the error rate or a slowing of the rate of decline), and the magnitude of the decrease also increases with the increase in the average photon number. Taken together, these results consistently indicate that the dissipation effect caused by atmospheric turbulence negatively impacts radar detection performance, reducing the radar's target detection capability.

[0124] 3.2 Influence of Phase Diffusion Channel

[0125] We calculated the performance of quantum illumination radar based on TMSV state when considering the influence of turbulent atmospheric phase diffusion channel.Figure 5 The trend of the von Neumann entropy is revealed. With the increase of the average number of emitted photons N emit , the von Neumann entropy E shows a gradual increasing trend. However, when the value of the phase diffusion intensity parameter Γ a increases (corresponding to stronger turbulence or longer transmission distance), the increase of these performance indicators is significantly reduced. In particular, the performance under high phase diffusion rate (larger Γ a value) is significantly worse than that under low phase diffusion rate or no phase diffusion (Γ a = 0). It is shown that the phase diffusion effect caused by turbulence also leads to the decline of the system detection performance, limiting the effective working range of quantum radar, especially under relatively high turbulence conditions, the limiting effect is more obvious.

[0126] As shown in Figure 6 , as the other two important indicators for measuring quantum detection performance, Helstrom error limit and QCB index also show similar trends. The increase of Helstrom error limit indicates that under the influence of turbulent phase diffusion, the theoretical best detection capability that quantum radar can achieve is significantly inhibited; at the same time, the change of QCB index (also manifested as the increase of error rate upper bound or the slowing down of the decrease rate) reflects the decrease of the system's tolerance to noise, resulting in the decrease of the system's robustness in complex environment.

[0127] 3.3 Detection performance analysis under high photon number

[0128] In the study of the change of Helstrom limit error probability with the average number of emitted photons under the influence of phase diffusion channel (as shown in Figure 6 (a)), we observed a phenomenon worthy of attention: with the increase of N emit , the decrease rate of P err,M shows a slowing down trend. In particular, for stronger phase diffusion intensity (for example, Γ a = 0.5), the downward trend even shows signs of saturation and flattening in the high N emit region. This preliminary observation prompted us to further explore the change law of detection performance in the high N emit interval. In order to systematically reveal the behavior characteristics in the high photon number domain, we performed extended range numerical simulation, and the results are shown in Figure 7 (a). At the same time, due to the increase of the upper limit of the number of photons involved in the calculation, we also increased the corresponding calculation truncation dimension, and plotted Figure 7 (b) to emphasize the convergence of the results of Figure 7 (a), which can fully prove that the calculation truncation dimension of Figure 7 (a) is reasonable.

[0129] Figure 7 (a) precisely depicts the influence of different phase diffusion strength Γ a (0.05, 0.1, 0.15, 0.2, 0.5) on the Helstrom limit error probability P err,M . Figure 7 (a) clearly shows that for all considered Γ a values, P err,M does not monotonically decrease with N emit , but rather exhibits a non-monotonic behavior: P err,M first decreases with N emit , reaches a minimum and then starts to increase with further increase of N emit . This means that for a specific Γ a , there exists an optimal average number of emitted photons N opt , for which P err,M reaches its minimum Moreover, a key trend is that with increasing phase diffusion strength Γ a , the optimal photon number N opt corresponding to the minimum P err,M moves towards lower values.

[0130] To quantify and systematically study the position N opt of the P err,M minimum and its corresponding minimum value as a function of the phase diffusion strength Γ a , we further extract the information of these minimum points, as shown in Figure 8 .

[0131] Figure 8 This clearly reveals the core dependence of how the system’s optimal working point changes with the phase diffusion strength Γ a . As shown in Figure 8 , the left axis exhibits a monotonically decreasing trend of the optimal average number of emitted photons N a required to reach the lowest error probability with increasing Γ opt , especially in the range of approximately 0-0.2. The results show that when the phase diffusion effect is enhanced, the system will reach its performance limit point at a relatively low level of emitted power. The existence of this performance limit point and its sensitivity to Γ a , especially when Γ a is small, are particularly pronounced. At the same time, the right axis Figure 8 also demonstrates how the system’s optimal performance itself is affected by Γ a : the minimum Helstrom limit error probability P err,M . monotonically increases with Γ a . It is clear that the increase of phase diffusion not only changes the optimal working point, but also fundamentally limits the optimal detection ability of the system. Even when working at the optimal photon number , stronger diffusion will lead to higher unavoidable error probability.

[0132] The numerical results reveal a key feature of quantum illumination radar performance in the presence of phase diffusion: for the system with TMSV states, increasing the transmitted power (increasing N emit ) is not always the optimal strategy. When N emit exceeds a certain threshold N opt , further increasing the photon number will not bring performance gain, but instead lead to performance reversal. The physical meaning of this phenomenon is that in the high photon number region, although the signal intensity increases, the destructive effect of phase diffusion on the signal also increases, and gradually becomes the dominant factor limiting the detection performance. The increase of signal photon number amplifies the effect of phase noise, and its negative effect eventually exceeds the positive benefit brought by signal enhancement. Figure 8 The results further quantify this physical picture: the stronger the phase diffusion (the larger Γ a ), the earlier the dominant role of noise appears (the smaller N opt ), and the worse the optimal performance the system can achieve (the larger ).

[0133] We quantitatively analyze the influence of atmospheric transmission effects on the key performance indicators of quantum radar (including von Neumann entropy, Helstrom error limit and QCB error limit) under different turbulence conditions using numerical simulation methods. The simulation results clearly show that atmospheric turbulence can significantly affect the transmitted quantum state, directly leading to the weakening of the detection ability of the radar system. In particular, under strong turbulence conditions, the performance decline is particularly pronounced.

[0134] Despite the adverse effects of turbulence, quantum illumination radar based on TMSV states still exhibits relatively strong anti-interference ability in turbulent atmospheric environments. Especially when mainly considering atmospheric dissipation effects, the system can still maintain a relatively low false alarm rate and missed alarm rate to some extent. However, in the case of significant phase diffusion effects, the decline in radar performance is more pronounced, indicating that phase noise is one of the key factors limiting its performance in strong turbulence or long-distance applications. In addition, the study also found a key phenomenon in the high power (high average photon number) region: there is an optimal average transmitted photon number N opt , which minimizes the error probability P err,M ; when N emit> N opt When the phase diffusion noise dominates, the performance is reversed. This finding emphasizes that it is not the best strategy to increase the transmitted power unconditionally in the actual turbulent environment, and a balance between signal enhancement and phase noise must be found.

[0135] Subsequent research should focus on revealing the specific physical mechanisms that lead to performance reversal, such as the contribution of high-order photon number terms to error probability under different noise models, and systematically comparing the performance differences of various noise models and turbulent conditions. On the other hand, these findings provide new ideas for quantum state parameter optimization. Given the advantages of neural networks in nonlinear modeling and high-dimensional optimization, deep learning methods can be used to automatically search and adjust quantum state transmission parameters. By constructing a multilayer perceptron network or a more complex deep network architecture suitable for quantum radars, the network can learn the comprehensive effects of thermal noise and atmospheric turbulence on quantum states from a large amount of simulation or experimental data, and optimize the outgoing state to maintain the best detection performance in complex environments. This method not only accurately simulates the effects of multiple sources of interference on quantum states, but also adaptively finds the global optimal solution in a multi-dimensional parameter space, thereby significantly improving the detection efficiency, environmental adaptability, and operational stability of quantum radar systems.

[0136] In summary, this study reveals the complex effects of turbulent atmosphere on the performance of TMSV state quantum illumination radar, providing valuable theoretical reference and performance evaluation basis for the design and deployment of quantum radar systems in the future. Future research directions can further explore how to optimize quantum light field design, improve receiving end measurement schemes, or combine adaptive optics and other compensation techniques to improve the detection performance of quantum radars in turbulent atmosphere, especially under high photon number and strong turbulence conditions, to promote the practical development of quantum radar technology.

Claims

1. A method for detecting turbulent atmospheric targets based on a dual-mode compressed vacuum state quantum illumination radar, characterized by: The steps include: S1. According to the working principle of quantum illumination radar, the TMSV state is selected as the emission light field of the quantum illumination radar, and the emission light field is described as the TMSV state density matrix ρ AB ; S2. Based on the influence of atmospheric turbulence on the propagation of the signal mode light field, a model is established to reveal the physical decoherence mechanism. The model is composed of the linear superposition of the atmospheric dissipation part described by the atmospheric dissipation rate and the phase diffusion part described by the phase diffusion rate. In the process of propagation of S3 and signal mode light field in turbulent atmosphere, the atmospheric dissipation part and the relationship between atmospheric dissipation rate and turbulent atmospheric transmittance are used to calculate the TMSV state density matrix ρ. AB Apply the Kraus operator to obtain the quantum state density matrix after loss Using the phase diffusion part and the dimensionless parameters estimated by Fried parameters and turbulent outer scale wave number, the TMSV state density matrix ρ AB Apply the Kraus operator to obtain the density matrix of the quantum state after diffusion S4. Model the target as having a transmittance of η t The beam splitter and the unitary operator U of the beam splitter BS Describe the interaction between the signal mode light field and the background thermal noise at the target, that is, the mixing process; S5. Assume that the area to be detected has two situations: the target exists and the target does not exist: If the target exists, the signal mode light field interacts with the background thermal noise at the target, and after being reflected to the receiving end, it produces a joint state ρ1 with the reference mode light field; If the target does not exist, only the background thermal noise is reflected to the receiver and produces a joint state ρ0 with the reference mode light field.

2. The method for detecting turbulent atmospheric targets based on a dual-mode compressed vacuum state quantum illumination radar according to claim 1, characterized in that: The TMSV state density matrix ρ AB It is obtained through the wave function of the TMSV state, and its expression is: AB =|ψ TMSV ><ψ TMSV |,|ψ TMSV > is the right arrow of the TMSV state.

3. The method for detecting turbulent atmospheric targets based on a dual-mode compressed vacuum state quantum illumination radar according to claim 2, characterized in that: The TMSV state is generated by the two-mode squeezing operator acting on the vacuum state, and its expression is: Where: |00> AB represents the vacuum state, and S2(r) represents the two-mode squeezing operator:

4. The method for detecting turbulent atmospheric targets based on a dual-mode compressed vacuum state quantum illumination radar according to claim 1, characterized in that: The influencing mechanisms include amplitude fluctuations and phase fluctuations.

5. The method for detecting turbulent atmospheric targets based on a dual-mode compressed vacuum state quantum illumination radar according to claim 1, characterized in that: The model is: Where: is the annihilation operator, is the photon number operator, γ A is the dissipation rate, Γ A is the phase diffusion rate.

6. The method for detecting turbulent atmospheric targets based on a dual-mode compressed vacuum state quantum illumination radar according to claim 1, characterized in that: The quantum state after the loss The expression is: Where: are the annihilation operators corresponding to the signal module A and the idle module B respectively.

7. The method for detecting turbulent atmospheric targets based on a dual-mode compressed vacuum state quantum illumination radar according to claim 6, characterized in that: The quantum state after the loss The derivation process: Consider the amplitude fluctuation and the resulting loss effect. The random inhomogeneity of the refractive index caused by atmospheric turbulence will distort the wavefront of the transmitted beam, resulting in a random light intensity distribution on the receiving plane, namely the atmospheric scintillation effect; this effect makes the transmittance of the atmospheric channel no longer a fixed value, but a random variable; for the transmitted quantum state, this is equivalent to experiencing a random loss channel; consider a dual-mode light field ρ AB In atmospheric transmission, the loss process can be described by the dual-mode form of the amplitude fluctuation part in model (5): Where: parameter γ A and γ B The relationship between τ and atmospheric transmittance T is as follows: For the atmospheric transmittance T of signal mode A A and the atmospheric transmittance T for the reference mode B B , when saved locally, T B =1, quantum state after loss Can be achieved by the initial state ρ AB The corresponding Kraus operator is: Afterwards you will get:

8. The method for detecting turbulent atmospheric targets based on a dual-mode compressed vacuum state quantum illumination radar according to claim 1, characterized in that: The unitary operator U BS The expression is: Where: and are the annihilation operators of the signal mode light field and the thermal noise mode thermal field, respectively.

9. The method for detecting turbulent atmospheric targets based on a dual-mode compressed vacuum state quantum illumination radar according to claim 1, characterized in that: The expression of the joint state ρ1 is: Where Tr C Perform partial trace operation on system C to extract subsystem information from the composite system density matrix; U BS The unitary operator describing the beam splitter transformation corresponds to the mixing and reflection of the light field at the target; The tensor product symbol is used to construct the density matrix of a composite system and describe the quantum state of a composite system composed of multiple subsystems; ρ′ AB is the density matrix of the two-mode entangled state after turbulence.

10. The method for detecting turbulent atmospheric targets based on a dual-mode compressed vacuum state quantum illumination radar according to claim 1, characterized in that: The expression of the joint state ρ0 is: Tr A Perform a partial trace operation on system A; The tensor product symbol is used to construct the density matrix of a composite system and describe the quantum state of a composite system composed of multiple subsystems; ρ′ AB is the density matrix of the two-mode entangled state after turbulence.

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