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Abstract
Quantum illumination and quantum entanglement provide an efficient method for detecting low-reflectivity targets in bright thermal-noise environments. Until now, almost all studies on quantum illumination assume that a target has only two mutually exclusive choices of being present and absent. In this study, we investigate the performance of quantum illumination when a target adopts a probabilistic mixed strategy. Remarkably, we show that the probabilistic mixed strategy can help the target hide its presence. This conclusion holds for asymptotic quantum illumination with joint measurement and practical quantum illumination with mode-by-mode local measurement. We also verify our result for Gaussian quantum illumination with a two-mode squeezed vacuum state and non-Gaussian quantum illumination with the NOON state. The mixed strategy is successful because of the large amount of resources consumed in target detection. This work is the next step toward better understanding of the stealth technique in future quantum illumination.
© 2022 Optica Publishing Group
1. INTRODUCTION
Quantum illumination (QI) [1–3], first proposed by Lloyd, has attracted significant interest. QI, i.e., quantum target detection, is a new quantum information technique for detecting low-reflectivity targets via quantum entanglement and correlation. Compared with classical illumination (CI), QI provides a higher signal-to-noise ratio and lower energy consumption. Existing studies on QI can be divided into three categories: (1) searching for more powerful quantum sources for target detection, e.g., QI with two-mode Gaussian [4] and non-Gaussian entanglement states [5–10]; (2) searching for efficient measurement devices. For example, Guha et al. designed a mode-by-mode optical receiver for practical target detection [11]. Sanz et al. developed an iterative method for efficient estimation in QI [12]. Zhuang et al. developed a distributed strategy to enhance the performance of quantum target detection [13,14]. Zhang et al. considered the target detection process as a communication task and proposed a displacement-then-antidisplacement method to enhance quantum target detection [15]. (3) The third is searching for all other possible applications. The principle of QI has been applied to eliminate eavesdropping in quantum communications [16–18] and for data reading in semiconductor devices [19]. In addition, QI methods have been applied to 3D quantum imaging [20] and quantum ranging [21]. QI experiments carried out in recent years have shown that illumination with a quantum protocol has clear advantages over CI [22–25].
However, in almost all of the above works, a target is assumed to have only two mutually exclusive choices of being present and absent. This leads to an intriguing question, i.e., what would be the target detection performance if a target adopts a probabilistic mixed strategy for the two choices. Surprisingly, our results show that such a target successfully escapes detection by QI systems. This can be considered as “stealth” in QI. Stealth is successful in QI systems because of a large amount of resources consumed in QI. To achieve a better-than-classic performance, QI requires a large number of entangled states (typically $M \gtrsim {10^6}$ pairs) to interrogate a target, which, if present, should passively and constantly reflect a signal. Thus, a target with the probabilistic mixed strategy may have higher chance of hiding [26–28] its presence. We verify our results with Gaussian state QI and non-Gaussian state QI.
The remainder of this paper is organized as follows. In Section 2, we describe CI with coherent states and QI with two-mode squeezed vacuum states (TMSSs). In Section 3, we briefly explain the probabilistic mixed strategy that helps hide a target. Section 4 presents the analysis of the working principle of a covert target with a 0–1 Fock state subspace. In Section 5, we consider a similar problem of practical QI with mode-by-mode local measurement. Section 6 describes the analysis of hiding targets in non-Gaussian state QI with the NOON state. Finally, we conclude the paper in Section 7.
2. TARGET DETECTION SCHEMES WITH MULTICOPY QUANTUM STATES
The schemes of CI and QI are shown in Figs. 1(a) and 1(b), respectively. In both schemes, a target is modeled with a beam splitter with transmittance ${T_0} = 1 - \kappa$ ($\kappa \ll 1$). When the target is absent, the signal is never reflected (corresponding to $\kappa = 0$) and the target-return signal is simply the thermal state, ${\rho _{\rm{E}}}$, in the ambient environment. In CI, the transmitter sends $M$ copies of independent and identical coherent states, $|\alpha \rangle$, whereas in QI, the transmitter prepares $M$ pairs of independent signal-idler quantum entanglement states (TMSS) and sends all signal modes to interrogate the target. Finally, joint quantum measurement is performed to extract the target information [29,30]. The problem of target detection can be mapped to a binary quantum hypothesis test in which the relevant quantum systems are prepared in $\rho _0^{\otimes M}$ under hypothesis ${H_0}$ (target absent) and in $\rho _1^{\otimes M}$ under hypothesis ${H_1}$ (target present). In general, $\rho _0^{\otimes M}$ and $\rho _1^{\otimes M}$ are nonorthogonal [31]. The minimal error probability for discrimination is given by
Fig. 1. Target detection with (a) coherent states and (b) two-mode squeezed vacuum states. Both schemes use a beam splitter with transmittance ${T_0} = 1 - \kappa$ to model the target and joint quantum measurement to extract the target information.
3. PROBABILISTIC MIXED STRATEGY
In this section, we show that a target can use a probabilistic mixed strategy to hide its presence. This is quite similar to covert communication, where messages are hidden in a large amount of innocuous noise [34]. For simplicity, we assume that a target, if present (hypothesis ${\tilde H_1}$), chooses a probabilistic mixed strategy such that the final state collected for joint measurement is given by $\tilde \rho _1^{\otimes M}$, with
Here, $q(0 \lt q \le 1)$ is the probability that the target chooses to reflect the interrogating signal. Such a mixed strategy can be implemented as follows: for each of $M$ independent interactions, the target chooses to reflect the signal with probability $q \ll 1$ and not reflect with probability $1 - q$. In other words, it randomly chooses $d = Mq$ out of $M$ modes of interrogating signals to reflect and leaves the other $M - d$ modes. In scenarios where $M \gg 1,d \ll M$, we show that, as more resources are consumed in target detection, it becomes easier for the target to hide its presence.In general, the evaluation of ${P_{{\rm{err}}}}$ is quite difficult for large $M$. Thus, the quantum Chernoff bound (QCB) [32,33] is used to find the corresponding lower and upper bounds: $P_{{\rm{err}}}^{\rm{L}} \lt {P_{{\rm{err}}}} \le P_{{\rm{err}}}^{\rm{U}}$, with
Fig. 2. Upper and lower bounds for error probability of quantum detection of (a), (c) conventional target and (b), (d) target with probabilistic mixed strategy. In (a) and (b), we consider a target with high reflectivity, $\kappa = 0.5$, environment with low noise, ${N_B} = 0.1$, and signal strength ${N_s} = 0.05$. All quantum states are truncated in a 2D photon subspace spanned by $\{|0\rangle ,|1\rangle \}$. In (c) and (d), we consider a low-reflectivity target hidden in a noisy environment with $\kappa = 0.1,{N_s} = 0.1,{N_B} = 0.5$. In (e) and (f), we consider the upper and lower bounds for the error probability of classic illumination: $\kappa = 0.1,{N_B} = 0.3,{N_s} = 0.1$. The quantum states in (c)–(f) are evaluated in the subspace spanned by $\{|0\rangle ,|1\rangle , \cdots ,|9\rangle \}$. The probability strategy in (b), (d), and (f) is $q = 1/M$.
It is interesting to investigate the value of error probability in the case of large values of $q$. In Fig. 3, we choose $q = d/M$ and find that the lower bound of the error probability still approaches the limit of $1/2$ for large values of $M$ and $d$. In Fig. 3(a), we consider a target with higher reflectivity, $\kappa = 0.5$, and other parameters are chosen to be ${N_s} = 0.05$ and ${N_B} = 0.1$. In this case, it is easily determined that $d_{{\rm{QI}}}^{{\max}} = 1.05 \times {10^3}$. The value of $d$ is chosen to be $d_{{\rm{QI}}}^{{\max}}/200,d_{{\rm{QI}}}^{{\max}}/100,d_{{\rm{QI}}}^{{\max}}/50,d_{{\rm{QI}}}^{{\max}}/20,d_{{\rm{QI}}}^{{\max}}/10$ from top to bottom. A larger value of $d$ means a slower speed at which the lower bound of the error probability approaches $1/2$. In Fig. 3(b), we consider a target with reflectivity $\kappa = 0.05$, ${N_B} = 0.1$, and signal strength ${N_s} = 0.05$, which follows $d_{{\rm{QI}}}^{{\max}} = 5.56 \times {10^3}$. The lower bound of the error probability behaves in a similar way to what is shown in Fig. 3(a).
Fig. 3. Lower bounds for error probability of quantum detection of a target with probabilistic mixed strategy. (a) $\kappa = 0.5,{N_s} = 0.05,{N_B} = 0.1,d_{{\rm{QI}}}^{{\max}} = 1.05 \times {10^3}$; (b) $\kappa = 0.05,{N_s} = 0.05,{N_B} = 0.1,d_{{\rm{QI}}}^{{\max}} = 5.56 \times {10^3}$.
4. BASIC WORKING PRINCIPLE OF THE HIDDEN TARGET
The basic working principle of the different behaviors in the detection of a conventional target and a target with the probabilistic mixed strategy can be clearly understood in the subspace spanned by Fock states $\{|0\rangle ,|1\rangle \}$. For a target with the probabilistic mixed strategy (hypothesis ${\tilde H_1}$), one obtains ${\tilde \rho _1} = \frac{1}{{1 + {\lambda ^2}}}\frac{1}{{1 +\mu}}\{(1 + q\mu \kappa)|00\rangle \langle 00| + q\lambda \sqrt \kappa (1 -\mu(1 - 2\kappa))$ $(|00\rangle \langle 11| + |11\rangle \langle 00|) +\mu(1 - q\kappa)|01\rangle \langle 01|\, + \,[q{\lambda ^2}(1 - \kappa)(1 + 2\mu \kappa) +$ $(1 - q){\lambda ^2}]|10\rangle \langle 10| + {\lambda ^2}[q(\kappa +\mu{(1 - 2\kappa)^2}) + (1 - q)\mu]|11\rangle \langle 11|\} ,\lambda = \sqrt {{N_s}/({N_s} + 1)} ,\mu = {N_B}/({N_B} + 1)$ and
Similarly, the lower bound for target detection with a coherent state has a result similar to $P_{{\rm{err,QI}}}^{\rm{L}}$, except that ${\delta _{{\rm{QI}}}}$ is replaced with ${\delta _{{\rm{CI}}}} = 2d\kappa (1 - \kappa){N_s}{N_B}/(({N_s} + 1)(2{N_B} + 1))$. The corresponding condition for escaping detection is $d \ll d_{{\rm{CI}}}^{{\max}} = \frac{{({N_s} + 1)(2{N_B} + 1)}}{{2\kappa (1 - \kappa){N_s}{N_B}}}$.
Consequently, a target with a smaller reflectivity ($\kappa$) is more suitable for hacking QI and CI systems with low signal strength (${N_s}$) in a low-noise environment (${N_B}$).
In order to conclude this section, we first provide two notes. The first is in regard to the comparison between the probabilistic mixed strategy and the fixed loss strategy. In the probabilistic mixed strategy, the final state received by the optical receiver is the state ${\rho _1}$, which is mixed by the state ${\rho _1}$ (target being present) with proportionality factor $q$ and the state ${\rho _0}$ (target being absent) with proportionality factor $1 - q$, as clearly shown in Eq. (2). Intuitively, there may exist a fixed loss strategy. The target may choose to decrease its reflectivity from $\kappa$ to $q\kappa$. In this case, the final quantum state received by the optical receiver will be ${\rho _1}$, where the reflectivity of the target will be $q\kappa$. When considering the received photon energy, these two strategies are the same. However, when considering the received quantum state, they are notably different. The lower bound of the error probability devised by Eq. (3) will be also different. We provide a plot of the lower bound of error probability in Fig. 4. Other curves such as the upper bounds are not provided, since the lower bound of error is sufficient enough to demonstrate whether the target can escape detection. If the lower bound approaches $1/2$, the error probability will be larger than $1/2$, and the target hides perfectly. We choose $\kappa = 0.5,{N_s} = 0.02,{N_B} = 0.5,q = d/M,d = d_{{\rm{QI}}}^{{\max}}/50,$ with $d_{{\rm{QI}}}^{{\max}} = 832$. For the probabilistic mixed strategy, the lower bound of error probability approaches $1/2$, whereas for the fixed loss strategy, the lower bound of error probability approaches 0.2960. Thus, the probabilistic mixed strategy is more efficient in hiding a target.
Fig. 4. Lower bound of error probability of probabilistic mixed strategy and of fixed loss strategy. Other parameters: $\kappa = 0.5,{N_s} = 0.02,{N_B} = 0.5,q = d/M,d = d_{{\rm{QI}}}^{{\max}}/50$.
The second note is about the multicopy and high-intensity quantum illumination. To enhance the performance of quantum illumination, the sender could increase the number $M$ of quantum entangled states or use a higher-intensity quantum entanglement with much higher average photon numbers. To make a fair comparison, we fix the average photon numbers that are sent to interact with the target. As has been well-established, in quantum illumination, only one mode of quantum entanglement, e.g., the B mode in Fig. 1, is sent to interact with the target. The average photon number in the B mode is ${N_s}$. Thus, in quantum illumination involving a factor of $M$ times the entanglement state preparation and transmission, a number of
photons are sent to interact with the target.Thus, by fixing ${M^\prime}$ and changing the value of $M$, one can study the performance of quantum illumination with multicopy and multivalues of intensity of the quantum entanglement state. In Figs. 5(a) and 5(b), we plot the lower bound of error probability in the case of $d = d_{{\rm{QI}}}^{{\max}}/20$ and $d = d_{{\rm{QI}}}^{{\max}}/100$, respectively. For a given average photon number ${M^\prime}$, we consider the quantum illumination with quantum entanglement states of ${N_s} = 0.05,0.10,0.15,0.20,0.25$. A higher value of ${N_s}$ means a smaller value of $M$, due to the fixed value of ${M^\prime}$. However, a larger value of ${N_s}$ is more favorable for quantum illumination, as it provides a smaller error probability. Taking $d = d_{{\rm{QI}}}^{{\max}}/20$ and ${M^\prime} = 5 \times {10^4}$, for example, the error probability for quantum illumination with ${N_s} = 0.05$ is $0.4923$, but for quantum illumination with ${N_s} = 0.25$ is only $0.4462$. This demonstrates that increasing the quantum entanglement is more efficient when compared with increasing the number of quantum entangled states in the quantum illumination. The generation of strong two-mode squeezed vacuum states is important, and, experimentally, squeezing of 10 dB (corresponding to ${N_s} = 1.98$) can be obtained in state-of-art laboratories [35].
Fig. 5. Lower bound of error probability in detecting a target adopting a probabilistic mixed strategy, with $\kappa = 0.5,{N_s} = 0.05,{N_B} = 0.1$ and $q = d/M$. Other parameters: (a) $d = d_{{\rm{QI}}}^{{\max}}/20$; (b) $d = d_{{\rm{QI}}}^{{\max}}/100$. Photons in each optical mode are simulated within 10-dimensional photon subspace.
5. HIDING IN PRACTICAL QUANTUM ILLUMINATION WITH MODE-BY-MODE MEASUREMENTS
In practice, it is difficult to implement the joint quantum measurement required to achieve the optimal error probability in the QCB. A room-temperature broadband quantum memory and large-scale quantum coherence are highly required. This has been a hot topic in recent years, and great efforts have been made to prolong the storage [36–39]. On the other hand, recent studies have shown a more practical method of implementing QI, in which one performs separable measurements over $M$ pairs of target-return and idler modes. In this section, we show that it is still possible to hide a target with the probabilistic mixed strategy in practical QI. QI with separable quantum measurements consists of the mode-by-mode measurement of $M$ copies of return-idler states. For each copy, the return and idler modes are input into an optic parametric amplifier (OPA). Then, the photon number of the return mode is counted by photon number resolving detectors.
Let ${n_i}(i = 1,2, \cdots M)$ be the photon number of the $i$th return mode. Then, the QI system decides in favor of hypothesis ${H_0}$ if $N = \sum\nolimits_{i = 1}^M {n_i}$ is below threshold value ${N_{{\rm{th}}}}$ and in favor of ${H_1}$, otherwise. For a target with only two mutually exclusive choices, it can be easily obtained that the return mode (after OPA) is a thermal state with mean photon number ${N_0}$ in the case of ${H_0}$(${N_1}$ in the case of ${H_1}$). Moreover, the probability density function of $N(N \ge 0)$ is
The threshold value is expressed as ${N_{{\rm{th}}}} = \frac{{M({\sigma _1}{N_0} + {\sigma _0}{N_1})}}{{{\sigma _0} + {\sigma _1}}}$. The error probability is ${P_{{\rm{err,OPA}}}} = {e^{- M{R_{{\rm{OPA}}}}}}/(2\sqrt {\pi M{R_{{\rm{OPA}}}}})({R_{{\rm{OPA}}}} = ({N_1} - {N_0}{)^2}/(2({\sigma _0} + {\sigma _1}{)^2}))$, which decreases exponentially with $M$. For a target with the probabilistic mixed strategy, the photon counts in the target-return mode (after OPA) follow the probability distribution given by
which can be approximated as a Gaussian distribution with mean ${\mu ^\prime} = d{N_1} + (M - d){N_0}$ and variance ${V^\prime} = d{N_1}(1 + {N_1}) + (M - d){N_0}(1 + {N_0})$. Hence, the error probability for detecting a target with the probabilistic mixed strategy is given byThis also holds for practical CI, where mode-by-mode homodyne measurement is applied for detection [11]. Let ${x_i}$ denote the Gaussian-distributed measurement result of each homodyne detection. An inference rule with the minimal error probability can be obtained by comparing the sum of all measurement results, ${X_{{\rm{mea}}}} = \sum\nolimits_{i = 1}^M {x_i}$, with a preassigned threshold value, ${X_{{\rm{th}}}}$: ${H_0}$ is declared if ${X_{{\rm{mea}}}} \lt {X_{{\rm{th}}}}$ and ${H_1}$, otherwise. For a target with only two mutually exclusive choices, it can be shown that ${X_{{\rm{mea}}}}$ is a Gaussian random variable that obeys ${X_{{\rm{mea}}}} \sim {\cal N}(M\sqrt {2\kappa {N_s}} ,M({N_B} + \frac{1}{2}))$ under ${H_1}$ (${X_{{\rm{mea}}}} \sim {\cal N}(0,M({N_B} + \frac{1}{2}))$ under ${H_0}$). The choice of ${X_{{\rm{th}}}} = M\sqrt {2\kappa {N_s}} /2$ induces an exponentially small error probability, ${P_{{\rm{err}}}} = \frac{1}{2}{\rm{erfc}}(\sqrt {M{R_C}}) \approx {e^{- M{R_C}}}/(2\sqrt {\pi M{R_C}})$, where ${R_C} = \kappa {N_s}/(4{N_B} + 2)$ [11]. Considerably different results may be obtained if a target with the probabilistic mixed strategy is present. For state ${\tilde \rho _1} = q{\rho _1} + (1 - q){\rho _0}$, the homodyne measurement of quadrature $x$ follows $\tilde p(x) = q{p_1}(x) + (1 - q){p_0}(x)$, where ${p_1}(x)$ and ${p_0}(x)$ are the probability densities of Gaussian random variables ${\cal N}(\sqrt {2\kappa {N_s}} ,{N_B} + \frac{1}{2})$ and ${\cal N}(0,{N_B} + \frac{1}{2})$, respectively. $\tilde p(x)$ is evidently non-Gaussian. According to the central limit theorem, measurement result ${X_{{\rm{mea}}}}$ follows a Gaussian distribution with mean $Mq\sqrt {2\kappa {N_s}}$ and variance $M{\tilde \sigma ^2}$, where $\tilde \sigma = \sqrt {2q(1 - q)\kappa {N_s} + ({N_B} + \frac{1}{2})}$. Thus, the lower bound for the error probability and its approximated value are given by
Fig. 6. Lower bound of error probability (lines marked with squares) and its approximated value (dotted lines) for detecting a target with probabilistic mixed strategy using mode-by-mode measurement in (a) quantum illumination and (b) coherent-state detection. Parameters used to generate the plots are $\kappa = 0.1,{N_s} = 0.05,{N_B} = 0.5$. The amplification gain is $G = 1 + {N_s}/\sqrt {{N_B}}$ in (a). All optic modes are truncated within an eight-dimensional subspace spanned by $\{|0\rangle ,|1\rangle , \cdots |7\rangle \}$.
6. HIDING IN NON-GAUSSIAN QUANTUM ILLUMINATION IN NOON STATE
In this section, we consider non-Gaussian QI with the NOON state as an example. The NOON state is a type of path-entangled state of $\aleph$ photons, and it plays an important role in quantum metrology [40] and quantum sensing [41]. The scheme of QI with the NOON state is the same as that shown in Fig. 1(b), except that entangled state $|\psi {\rangle _{{{\rm{A}}_{\rm{i}}}{{\rm{B}}_{\rm{i}}}}}$ is replaced withthe NOON state $|{\psi _{{\rm{NOON}}}}\rangle = \frac{1}{{\sqrt 2}}{(|\aleph \rangle _{{{\rm{A}}_{\rm{i}}}}}|0{\rangle _{{{\rm{B}}_{\rm{i}}}}} + |0{\rangle _{{{\rm{A}}_{\rm{i}}}}}|\aleph {\rangle _{{{\rm{B}}_{\rm{i}}}}})$. For convenience, we assume ${N_B} = 0$ to clearly show the working principle. Using a similar method, we obtain quantum state ${\rho _1}$ when the target is present and ${\rho _0}$ when it is absent: ${\rho _1} = \frac{1}{2}{[|\aleph \rangle _{{{\rm{A}}_{\rm{i}}}}}\langle \aleph | \otimes |0\rangle _{{{\rm{E}}_{\rm{i}}}}\langle 0| + {\kappa ^{\aleph /2}}(|\aleph 0\rangle _{{{\rm{A}}_{\rm{i}}}{{\rm{E}}_{\rm{i}}}}\langle 0\aleph | + |0\aleph \rangle _{{{\rm{A}}_{\rm{i}}}{{\rm{E}}_{\rm{i}}}}\langle \aleph 0|) \,+ |0\rangle _{{{\rm{A}}_{\rm{i}}}}\!\langle 0| \otimes \sum\nolimits_{m = 0}^\aleph \!\left(\!\begin{array}{*{20}{c}}{\aleph}\\m\end{array}\!\!\right){\kappa ^m}{(1 - \kappa)^{\aleph - m}}|m\rangle _{{{\rm{E}}_{\rm{i}}}}\langle m|],{\rho _0} = \frac{1}{2}(|\aleph \rangle _{{{\rm{A}}_{\rm{i}}}}{\langle \aleph | + |0\rangle _{{{\rm{A}}_{\rm{i}}}}}\langle 0|) \otimes |0{\rangle _{{{\rm{E}}_{\rm{i}}}}}\langle 0|.$ For a target with non-Gaussian QI, we have
In Fig. 7, we compare the exact and approximated values of ${\rm{Tr}}[\rho _0^{1/2}\tilde \rho _1^{1/2}]$ and $P_{{\rm{err,NOON}}}^{{\rm{L,app}}}$ with their numerical values for $\aleph = 5,q = 1/M,\kappa = 0.05$. Numerical simulation provides clear evidence of the $1/M$ scaling of $P_{{\rm{err,NOON}}}^{{\rm{L,app}}}$ for increasing $M$. For $M = 50 \gg 1$, the approximated value is $P_{{\rm{err,NOON}}}^{{\rm{L,app}}} = 0.3355$, which is a good approximation of the numerical value, $P_{{\rm{err,NOON}}}^{\rm{L}} = 0.3350$. Another phenomenon shown in Fig. 7 is that $P_{{\rm{err,NOON}}}^{{\rm{L,lim}}}$ is strictly smaller than ${P_{{\rm{err}}}} = \frac{1}{2}$ in the hacking of TMSS-based QI. Nevertheless, the nonzero value of the lower bound, $P_{{\rm{err,NOON}}}^{{\rm{L,app}}}$, is sufficient to show that more copies of entanglements cannot induce an exponentially smaller error probability and NOON state QI can be hacked.
Fig. 7. (a) Exact and approximated values of ${\rm{Tr}}[\rho _0^{1/2}\tilde \rho _1^{1/2}]$ in NOON state quantum illumination. (b) Exact and approximated values, $P_{{\rm{err,NOON}}}^{{\rm{L,app}}}$, of the lower bound of the error probability as a function of the number of entangled states, $M$. Black solid line indicates the limiting value, $P_{{\rm{err,NOON}}}^{{\rm{L,lim}}}$. Other parameters: $q = 1/M,{N_B} = 0,\kappa = 0.05,\aleph = 5$.
7. CONCLUSIONS
In this work, we have demonstrated the performance of QI for a target with a probabilistic mixed strategy. However, further work is required in order to make this more easily implementable. Here, we have provided two notes:
- (1) For a microscopic target, such as a photon or electron, the mixed strategy is relatively easy. We know that they induce quantum effects and that they can be in a superposition state of being in two places at the same time. Examples can be found in Young’s double-slit experiment. Moreover, recent theoretical and experimental work shows that quantum devices can be easily prepared in a superposition state. For example, a “quantum” beam splitter in the superposition state of being present or absent is proposed to analyze a quantum version of Wheeler’s delayed-choice experiment [42]. In recent years, these quantum beam splitters have been experimentally validated in laboratories with polarization entanglement-enabled interferometers [43], reconfigurable integrated quantum photonic circuits [44], single photons in an interferometer [45], a liquid ensemble in nuclear magnetic resonance [46], and even superconducting quantum circuits [47]. Recent studies have shown that the mechanical resonator could also exhibit the superposition effect, of which more information can be found in [48].
- (2) For a macroscopic target, such as a plane, it is relatively hard to induce into a quantum superposition state. However, there are some suggestions for doing so. Properties of the macroscopic target could be changed, such as its reflectivity and shape, or moving the direction to change its reflectivity during the radar’s $M$-times interrogation. As we demonstrate, if it exists that $d \ll d_{{\rm{QI}}}^{{\max}}$; then, in terms of the ensemble average, its state can be described by Eq. (2), and it can escape from being detected. This means a target with a superfast velocity can still potentially escape from being detected, even in the presence of quantum radar.
The success of escaping detection relies on the large number of entangled states required in QI. Mathematically, this is due to the exponential limit, $ \lim\limits_{M\to \infty }\,{{(1-\frac{d\epsilon }{M})}^{M}}={{e}^{-d\epsilon }}\gt 0$, for all values of $d \lt M$. For $d = M$, the exponential limit vanishes, and the probabilistic attacking method automatically fails. Hence, QI with less entanglement must be developed to implement quantum radars against stealth targets. In particular, a one-shot QI scheme with $M = 1$ is inherently immune to probabilistic stealth targets. The security and sensitivity of QI systems can be enhanced if parameters ${N_B},M$, and ${N_s}$ are maintained as private in QI with a TMSS ($\aleph$ in the NOON state). Without these parameters, a stealth intruder will have no information on ${d_{{\max}}}$ and would have to decreases its probability ($q$) to escape detection. Our work demonstrates one of the possible methods for hiding a target in state-of-the-art QI. We hope that such a method can be verified through QI experiments.
Funding
National Natural Science Foundation of China (11981240356, 62171036); Beijing Institute of Technology Research Fund Program for Young Scholars.
Acknowledgment
The authors thank Prof. Song Yang for fruitful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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