李旭, 聂敏, 杨光, 裴昌幸. 基于纠缠的量子雷达生存性策略及性能仿真. 2015, 44(11): 1127002-1127006
LI Xu, NIE Min, YANG Guang, PEI Chang-xing. The Strategy and Performance Simulation of Quantum Entangled Radar's Survivability. ACTA PHOTONICA SINICA, 2015, 44(11): 1127002-1127006  Permissions基于纠缠的量子雷达生存性策略及性能仿真李旭¹, 聂敏¹, 杨光^1,2, 裴昌幸³摘要关键词: 量子通信; 量子纠缠; 保真度; 雷达; 目标探测中图分类号:O413.3;O413.1 文献标志码:A 文章编号:1004-4213(2015)11-1127002-5The Strategy and Performance Simulation of Quantum Entangled Radar's SurvivabilityLI Xu¹, NIE Min¹, YANG Guang^1,2, PEI Chang-xing³AbstractKeyword: Quantum communication; Quantum entanglement; Fidelity; Radar; Target detectionShow Figures0 引言

电子对抗技术和隐身技术的发展和应用, 使经典雷达探测受到了挑战.因此, 研发一种反隐身、准确度高、探测范围广、抗电磁干扰的新一代雷达成为了研究热点^[1].基于纠缠的量子雷达可以增强探测能力^{[2, 3, 4, 5]}, 并可实现对隐身平台的探测.2008年美国麻省理工学院的Lloyd教授首次提出了量子远程探测系统模型— 量子照射雷达^[6], 从理论上证明了量子力学可以应用于远程目标探测; 2009年中国科学院的刘伍明团队^[7, 8]对量子纠缠系统的演化进行了深入研究, 为量子纠缠动力学的实验研究提供了理论依据; 2012年中国科技大学的潘建伟团队首次实现了百公里级的自由空间量子隐形传态和纠缠分发^[9], 为远距离量子探测奠定了基础; 2013年, 意大利的Lopaeva 等^[10]首次用实验方法实现了量子照射雷达, 该实验基于光子数关联^[11], 验证了Lloyd 提出的量子照射雷达模型探测在高噪声及高损耗时依然有目标探测能力; 2015年, 德国亚琛工业大学的Shabir Barzanjeh等^[12]对微波量子照明探测进行了深入研究.

环境对量子态传输的干扰是研制高品质量子雷达需要克服的障碍之一^[13, 14], 量子雷达需要具有在复杂环境下的生存能力和工作能力.为此, 本文提出一种分析量子雷达生存性的方案, 首先分析了探测光子的保真度, 并建立探测光子的生存函数, 然后定义了干扰等级, 并根据不同干扰等级下的量子损伤模型建立量子雷达生存函数, 最后通过对量子雷达和探测光子的生存函数的仿真, 分析不同程度的量子干扰对量子雷达生存性的影响.

1 量子雷达探测原理

根据量子雷达发射机发射量子态的不同, 量子雷达可分为单光子量子雷达和纠缠光子量子雷达.与不采用纠缠的量子雷达相比, 采用纠缠的量子雷达分辨率以二次方速率提高^[15].

基于纠缠的量子雷达, 其探测过程如图1.利用泵浦光子穿过(BBO)晶体, 通过参量下变换产生大量纠缠光子对, 各纠缠光子对之间的偏振态彼此正交, 将纠缠的光子对分为探测光子和成像光子, 成像光子保留在量子存储器中^[16], 探测光子由发射机发射经目标反射后, 被量子雷达接收, 根据探测光子和成像光子的纠缠关联可提高雷达的探测性能.

探测过程以一对纠缠的探测光子s和成像光子a为例, 两光子在自由度的角度动量上纠缠态为

|ψ > ^(e)= 1d∑k=1d" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">1d√∑k=1d$\begin{array}{c}\frac{1}{\sqrt[]{d}}\end{array}\begin{array}{c}\stackrel{d}{\sum _{k=1}}\end{array}$|k> _s|k> _a (1)

式中|k> 代表k模式的单光子态, |Ψ > ^(e)代表一个双光子系统, 求得纠缠态密度矩阵为

ρ ^e=|Ψ > ^(e)(e)< Ψ |(2)

当探测空间没有目标时, 探测光子丢失, 检测到的光子为噪声光子, 并以平均光子数b进入到系统的每个模式下, 每个模式的量子态为

ρ ₁≈ (1-db)|0> < 0|+b ∑k=1d" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">∑k=1d$\begin{array}{c}\stackrel{d}{\sum _{k=1}}\end{array}$|k> < k|(3)

式中|0> 代表真空态, 此时成像光子变为完全混合态, 即

ρ1e" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ρe1$\begin{array}{c}{\rho }_1^e\end{array}$≈ ρ ₁⊗ Iad" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Iad$\begin{array}{c}\frac{I_a}{d}\end{array}$+Ο (b²)≈ [(1-db)|0> _s< 0|+bI_s]⊗

Iad" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Iad$\begin{array}{c}\frac{I_a}{d}\end{array}$+Ο (b²)(4)

式中I_s和I_a是探测光子和成像光子在单光子Hilbert空间上的恒等算符, 有

I_X= ∑k=1d" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">∑k=1d$\begin{array}{c}\stackrel{d}{\sum _{k=1}}\end{array}$|k> _X< k|(5)

其中X=s或a.

探测区有目标时系统探测到初始纠缠态的概率为p_o, 探测到噪声光子 ρ1e" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ρe1$\begin{array}{c}{\rho }_1^e\end{array}$的概率为(1-p_o), 探测光子将被热化, 成像光子量子态可以用原始态来表示, 即

ρ2e" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ρe2$\begin{array}{c}{\rho }_2^e\end{array}$=(1-η ) ρ1e" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ρe1$\begin{array}{c}{\rho }_1^e\end{array}$+p_oρ ^e (6)

2 探测光子的生存性分析

量子雷达采用纠缠光子进行空间探测, 在探测过程中, 不可避免地会与探测环境相互作用, 从而产生纠缠, 影响探测结果, 且由于损耗存在, 光子会丢失或被信道吸收.根据复杂环境下量子保真度, 和光子生存系数η , 定义量子雷达探测量子信号的生存性, 若经过探测空间后探测信号的量子态与初始量子态越接近, 就说明探测量子信号的生存性越好, 量子雷达的探测能力更强.

2.1 探测光子生存函数的定义

设探测环境的信道运算元为

E₀= 1001-p" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">[1001−p−−−−√]$\begin{array}{c}\left[\begin{array}{cc}1& 0\\ 0& \sqrt[]{1-p}\end{array}\right]\end{array}$, E₁= 0p00" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">[00p√0]$\begin{array}{c}\left[\begin{array}{cc}0& \sqrt[]{p}\\ 0& 0\end{array}\right]\end{array}$(7)

式中p是一个光子丢失的概率, 一个光子的丢失相当于E₁把|1> 状态变为|0> 状态

假设量子雷达中一个量子比特状态描述为

ρ = ρ00ρ01ρ10ρ11" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(ρ00ρ10ρ01ρ11)$\begin{array}{c}\left(\begin{array}{cc}{\rho }_{00}& {\rho }_{01}\\ {\rho }_{10}& {\rho }_{11}\end{array}\right)\end{array}$(8)

探测过程中, 在t₀+Δ t时刻量子的密度算符描述为

ρ _E(t₀+Δ t)= ∑p=01" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">∑p=01$\begin{array}{c}\stackrel{1}{\sum _{p=0}}\end{array}$E_pρ E^† _p= ρ00+pρ111-pρ011-pρ10(1-p)ρ11" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(ρ00+pρ111−p−−−−√ρ101−p−−−−√ρ01(1−p)ρ11)$\begin{array}{c}\left(\begin{array}{cc}{\rho }_{00}+p{\rho }_{11}& \sqrt[]{1-p}{\rho }_{01}\\ \sqrt[]{1-p}{\rho }_{10}& (1-p){\rho }_{11}\end{array}\right)\end{array}$(9)

利用数学归纳法可得到在t₀+nΔ t时刻密度算符为

ρ _E(t₀+nΔ t)= ρ00+1-(1-p)nρ11(1-p)nρ01(1-p)nρ10(1-p)nρ11" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(ρ00+[1−(1−p)n]ρ11(1−p−−−−√)nρ10(1−p−−−−√)nρ01(1−p)nρ11)$\begin{array}{c}\left(\begin{array}{cc}{\rho }_{00}+\left[1-{(1-p)}^n\right]{\rho }_{11}& {\left(\sqrt[]{1-p}\right)}^n{\rho }_{01}\\ {\left(\sqrt[]{1-p}\right)}^n{\rho }_{10}& {(1-p)}^n{\rho }_{11}\end{array}\right)\end{array}$(10)

若量子雷达发射光子的初始量子态为

|φ > _Q=g|0> _E+h|1> _E(11)

则Q的密度算符可表示为^[17]

ρ _Q= g2ghghh2" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(|g|2ghgh|h|2)$\begin{array}{c}\left(\begin{array}{cc}{\left|g\right|}^2& gh\\ gh& {\left|h\right|}^2\end{array}\right)\end{array}$(12)

由纯态和任意量子态的保真度公式, 可得量子雷达发射的量子态经过复杂探测环境在t₀+nΔ t时刻的保真度为

F_c= <φ|Qρ|φ>Q" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;"><φ|Qρ|φ>Q−−−−−−−−−−−√$\begin{array}{c}\sqrt[]{<\phi {|}_Q\rho |\phi {>}_Q}\end{array}$=

g4+g2h2-g2h2(1-p)n+2g2h2(1-p)n+(1-p)nh4" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">g4+g2h2−g2h2(1−p)n+2g2h2(1−p−−−−√)n+(1−p)nh4−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√$\begin{array}{c}\sqrt[]{g^4+g^2{h}^2-g^2{h}^2{(1-p)}^n+2g^2{h}^2{\left(\sqrt[]{1-p}\right)}^n+{(1-p)}^n{h}^4}\end{array}$(13)

其中ρ 表示任意态密度算符.

令g²=μ , h²=ν , ( 1-p" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">1−p−−−−√$\begin{array}{c}\sqrt[]{1-p}\end{array}$)ⁿ=r, 由|g|²+ h2" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">|h|2$\begin{array}{c}{\left|h\right|}^2\end{array}$=1可将式(13)进一步化简为

F_c= μ2r(1-μ)(1-r(1-1/2μ))+1]" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">μ[2r(1−μ)(1−r(1−1/2μ))+1]−−−−−−−−−−−−−−−−−−−−−−−−−−√$\begin{array}{c}\sqrt[]{\mu \left[2\right.r(1-\mu )(1-r(1-1/2\mu \left)\right)+1]}\end{array}$(14)

则量子雷达的探测光子在干扰环境下的生存函数可表示为

Y_c=η F_c=η μ2r(1-μ)(1-r(1-1/2μ))+1]" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">μ[2r(1−μ)(1−r(1−1/2μ))+1]−−−−−−−−−−−−−−−−−−−−−−−−−−√$\begin{array}{c}\sqrt[]{\mu \left[2\right.r(1-\mu )(1-r(1-1/2\mu \left)\right)+1]}\end{array}$(15)

其中0< η < 1, 表示光子的生存系数.

2.2 仿真分析

若初始量子态为|φ > _Q= 12" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">12√$\begin{array}{c}\frac{1}{\sqrt[]{2}}\end{array}$|0> _E+ 12" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">12√$\begin{array}{c}\frac{1}{\sqrt[]{2}}\end{array}$|1> _E, 则在不同条件下探测光子的生存函数Y_c, 与探测时隙次数n的关系仿真如图2.

图2中, 量子生存系数η =0.9, 从图中可以看出, 探测光子的生存函数随着探测时隙次数的增加而降低, 说明探测光子探测时间越长, 生存性越低即损伤越严重, 当n→ ∞ 时即探测时间趋于无穷时, 生存性收敛与η μ" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">μ√$\begin{array}{c}\sqrt[]{\mu }\end{array}$=0.9· a=0.6363; 且探测光子生存性随着p的增大也会减小.图3中, 若光子丢失概率p=0.01, 从图中可以看出随着探测时间的增加探测光子生存性收敛于η μ" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">μ√$\begin{array}{c}\sqrt[]{\mu }\end{array}$, 而量子生存系数高的探测光子的生存性比量子生存系数低的高, 说明提高复杂环境下量子子生存系数可以整体提高探测光子的生存性, 从而增大量子雷达的探测距离.

3 量子雷达的生存性

在量子雷达探测过程中, 会受到扬沙、雾霾、PM2.5等恶劣天气的影响, 而且敌对方也可能会对量子雷达发出较强的量子干扰, 在设计和使用时需考虑它的抗干扰能力和生存能力, 因此提出量子雷达的生存性.

3.1 量子雷达生存函数的定义

量子雷达的生存性指在不同量子干扰条件下, 量子雷达仍然能够探测到目标的概率.概率越大, 该量子雷达的探测性越好, 信息传输越可靠, 抵抗干扰的能力越强.

为了描述不同干扰环境对量子雷达干扰的强度, 定义干扰等级α , 引入量子雷达信息保真度F_α , 它表示在干扰等级为α 时, 量子接收机接收到量子信息的保真度, 它们关系定义为

Fα=0.99,α=0Fα=χαFα-1,0<χ<1,1≤α≤10" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">{Fα=0.99,Fα=χαFα−1,0<χ<1,α=01≤α≤10$\begin{array}{c}\left\{\begin{array}{ccc}{F}_{\alpha }=0.99,& & \alpha =0\\ F_{\alpha }={\chi }^{\alpha }{F}_{\alpha -1},& 0<\chi <1,& 1\le \alpha \le 10\end{array}\right.\end{array}$(16)

根据式(16)可知, 在α =0的理想条件下, 量子雷达的量子信息保真度F₀为0.99.随着干扰等级α 的增大, 量子雷达的信息保真度为低一等级保真度的χ ^α 指数衰减, 且干扰等级越高, 衰减速度越快.

量子雷达发射的探测光子经目标反射后, 量子接收机接收返回的光子, 这一过程中平均损伤的量子比特数为N_m, 当信息保真度小于0.9时, 称之为损伤的量子态.假定在量子雷达探测中, 量子传输的总长度为L, 量子损伤传输概率为^[18]

P_ω =1-2 min(λ,ω)2" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">|min(λ,ω)|2$\begin{array}{c}{\left|\mathit{min}(\lambda ,\omega )\right|}^2\end{array}$(1-B)(1-p_λ )(17)

式中B为阻塞率, p_λ 为经典信道系统误码率, λ 、ω 为量子纠缠信道系数.传输长度L的平均损伤量子态个数N_m为

N_m=P_ω · L(18)

则接收机接收到一次量子态的平均保真度为

F_r= ∑i=1NmFiNm" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">∑i=1NmFiNm$\begin{array}{c}\frac{\stackrel{N_m}{\sum _{i=1}}{F}_i}{N_m}\end{array}$(19)

式中F_i= <φ|ρ|φ>" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;"><φ|ρ|φ>−−−−−−−−√$\begin{array}{c}\sqrt[]{<\phi \left|\rho \right|\phi >}\end{array}$为第i个量子态的保真度, < φ |表示量子态, ρ 表示量子态密度算符, 此时的F_r为量子雷达进行一次探测的量子信息保真度.根据干扰等级α , 假设探测过程中量子损伤个数m服从泊松分布, 即

P(m)=e^-α · αmm!" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">αmm!$\begin{array}{c}\frac{{\alpha }^m}{m!}\end{array}$(20)

在量子雷达系统中, 规定探测过程中量子损伤个数m达到损伤上限个数N_T时, 该量子雷达无法成功探测目标.设目标飞行体表面被探测点数为x, 结合探测光子在复杂环境下的生存性, 量子雷达的生存函数表示为

Y_r= ∑i=0xxi∑m=NT∞P(m)i1-∑m=NT∞P(m)x-i" role="presentation" style="margin: 0px; padding: 0px; display: inline; line-height: normal; text-indent: 0px; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">∑i=0x(xi)[∑m=NT∞P(m)]i[1−∑m=NT∞P(m)]x−i$\begin{array}{c}\stackrel{x}{\sum _{i=0}}\end{array}\begin{array}{c}\left(\begin{array}{c}x\\ i\end{array}\right)\end{array}\begin{array}{c}{\left[\stackrel{\infty }{\sum _{m=N_T}}P\left(m\right)\right]}^i\end{array}\begin{array}{c}{\left[1-\stackrel{\infty }{\sum _{m=N_T}}P\left(m\right)\right]}^{x-i}\end{array}$· Y_c (21)

根据量子雷达生存函数, 可以更好地研究各因素对量子雷达生存性的影响.

3.2 仿真分析

根据式(21), 量子态的干扰等级α 、目标飞行体表面被探测点数为x、量子生存系数η 及量子损伤上限N_T, 对量子雷达生存函数Y_r的影响如图4.

图4中, N_T=10, Y_c设为0.9, 从图中可以看出, 随着目标飞行体表面可探测点数的增加, 量子雷达的生存性越高, 探测能力越强.当干扰等级α =2, x=40时, 量子雷达的生存函数为0.9; 当x确定, α =6时, 生存函数降低到0.5, 说明环境中的干扰对量子雷达生存函数影响较大, 应适当调整量子雷达的设置参量以降低干扰和较小目标可探测面积对量子雷达生存性的影响.

当 x=30, p设为0.01, 干扰等级α =5, n=500时, 光子生存系数η 及量子损伤上限个数N_T与量子雷达生存函数Y_r的关系如图5, 从图中可以看出量子雷达生存函数随着量子生存系数的降低而减少, 在量子损伤上限为2条件下, 当η =0.9时, 生存函数可达0.9, 当η =0.5时, 生存函数降低到0.6.当η 固定时生存函数在N_T=2处取值最大.因此, 设定适当的量子雷达的量子损伤上限, 并提高量子在复杂环境下的生存性, 可提高量子雷达的生存性.

4 结论

为了判断量子雷达在不同干扰背景下的生存性, 本文定义了探测光子信号的生存函数, 基于干扰程度不同的量子损伤模型及探测光子信号生存性, 建立了量子雷达的生存函数, 并进行仿真.分析结果表明量子雷达的生存性随着量子干扰等级的增加而降低, 与理论相符.因此, 本文提出的量子雷达的生存函数可以作为一种研究量子雷达生存性的依据, 结合各种量子干扰对量子传输的影响, 设定适当的参量可以提高量子雷达的生存性和目标探测能力.

The authors have declared that no competing interests exist.

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