1. Introduction

Quantum entanglement is the most distinctive phenomena of quantum physics, which can lead to quantum enhancement in information technology [1]. In addition to the well-known quantum technologies such as quantum communication [2], quantum computing [3,4] and quantum imaging [5], quantum radar has also attracted much attention in recent years [6–11]. Quantum illumination (QI) is considered as the most promising quantum sensing protocol [12–14], and it has been experimentally demonstrated that quantum radar has the capabilities beyond the optimal classical radar in specific scenarios [15,16]. It is of great practical significance that QI can still provide quantum advantages in high noise and high loss links even when the entanglement is broken [17]. The reason is that the residual quantum correlation in entangled-state will not disappear with the destruction of entanglement.

In 2008, Tan et. al. showed that QI’s error-probability exponent is 6 dB higher than that of the optimal classical illumination (CI) through the quantum Chernoff bound (QCB) derived based on Bayesian method [18]. Based on the Neyman-Pearson (NP) criterion, Zhuang et. al. proved that QI has a theoretical gain of 6 dB than the optimal CI by analyzing the receiver operating characteristic (ROC) in 2017 [19]. Subsequently, in 2018, Giacomo showed that two-mode squeezed vacuum (TMSV) state is the optimal probe for QI in the scenario of asymmetric discrimination, and it can offer 6 dB gain in the error-probability exponent compared to the optimal CI [20].

However, it is difficult to construct a receiver that can perform optimal quantum information joint processing in practice, which may require the use of quantum computers. At present, the best QI receiver scheme proposed by Zhuang, namely feed-forward sum frequency generation (FF-SFG) receiver, can only approach the QI’s QCB in theory, and has not been realized in engineering [21]. Fortunately, two practical receiver schemes have been proposed by Guha, namely optical parametric amplification (OPA) receiver and phase conjugation (PC) receiver [22]. Both of two receivers are based on classical and realizable technologies, and they can theoretically achieve partial quantum enhancement in error-probability exponent, which is about 3dB. The implementation of these two receivers provides strong support for the experimental research of QI [23]. For instance, in 2015, Zhang et. al. experimentally demonstrated an entanglement-enhanced quantum sensing system with OPA receiver. In this experiment, signal-to-noise ratio (SNR) is defined as the ratio between the signal peak and noise background, which are represented by the 0 phase-modulated probe and $\pi$ phase-modulated probe respectively. The measured data shows that the implemented system realizes a 20% improvement of the output SNR over the optimal CI in an entanglement-breaking environment [24]. In 2020, Barzanjeh et. al. carried out experimental verification of QI in microwave band by constructing a digital PC receiver, and showed that it has 1 dB advantage over the optimal CI [25].

As we know, one of the purposes of signal processing is to improve the SNR of the receiver. This improvement obtained by signal processing, called SNR gain, is an important index to measure the performance of radar. Both transmitting sources and receivers will affect the SNR gain of radar system. For classical radar, matched filter (MF) is the most popular signal processing method, which is also considered as the optimal signal processing method under white noise in classical radar theory. However, the theoretical research on the quantum enhancement in SNR gain obtained by signal processing for QI radar has not been reported. QCB and ROC used in recent theoretical research to characterize the advantages of QI radar [18–20] are mainly based on the concept of error-probability exponent, which are derived from the distinguishability between two quantum states, but the error-probability exponent is not a recognized figure of merit in the radar literature, nor does it correspond in a straightforward manner to any such figure of merit. What’s more, the relevant experimental researches only make measurement or inference on the output SNR of the receiver without systematic theoretical analysis [26]. As discussed before, considering that the SNR gain obtained by signal processing is an important index in radar filed, which directly reflects the sensitivity of the radar, it is necessary to study the theoretical enhancement of SNR gain for QI radar. In this paper, we first compare MF with PC receiver, then define cases ${{\rm{H}}_{\rm{0}}}$ and ${{\rm{H}}_{\rm{1}}}$ from the binary detection problem in radar detection theory, and calculate the output of OPA/PC receiver in QI under both cases. Further, we theoretically analyze the SNR gain obtained by signal processing of QI and CI. Based on these analyses, we comprehensively study the quantum enhancement of QI in signal processing.

2. Physical fundamental of MF and PC receiver

In classical radar theory, pulse compression or MF can increase the SNR by $WT$ times, where W is the bandwidth of the signal, and T is the time-width of a pulse or the duration of one detection event for continuous signals. Assume a constant amplitude baseband signal ${A_S}\exp [j{\theta _S}(t)]$, then the MF operation can be expressed as

In Eq. (1), ${A_I}\exp [j{\theta _I}(t)]$ is the copy of the signal, which called idler. $\bar A_I^{}\exp [ - j{\theta _I}( - t)]$ is the impulse response of MF, where $\bar A_I^{}$ is the complex conjugate of ${A_I}$. Symbol ‘${\ast} $’ denotes the convolution operator, and ‘${\otimes}$’ denotes the cross-correlation operator. It should be noted that in classical physics, the idler has the largest classical correlation with the signal, but no quantum correlation.

If we consider the quantum case, the position operator $\hat q(t)$ and the momentum operator $\hat p(t)$ in quantum signal correspond to the in-phase component $I(t)$ and quadrature component $Q(t)$ in classical signal, respectively. Hence the classical complex signal $S(t) = {A_S}\exp [j{\theta _S}(t)] = {I_S}(t) + {Q_S}(t)$ is related to the creation operator $\hat a_S^\dagger (t) = \hat q_S^\dagger (t) - i\hat p_S^\dagger (t)$ in the quantum mode. Therefore, the result of MF for one mode can be preliminary expressed as

It should be mentioned that homodyne detection is the best receiving method for coherent-state and it can be regarded as the optimal classical receiver. It has advantages over PC receiver when the signal power is very small. Further, we can infer that MF cannot reach the signal processing gain ${G_c} = WT$ defined in the classical theory, and it only can approach to it if the signal is strong enough and the quantum noise can be ignored. Generally, the defined quantum enhancement only refers to the part beyond the optimal performance allowed by classical physics. Therefore, quantum enhancement only represents the gain when QI is compared with the optimal CI (homodyne detection for the coherent state) for quantum radar. It has been proved theoretically and experimentally that this enhancement in QI exists. So, how is it reflected in radar signal processing?

3. Analysis of the output mode for practical QI receivers

3.1 Two- mode compressed vacuum state and classical signal

First of all, consider the TMSV signal generated by spontaneous parametric down conversion (SPDC) in light band [27] or Josephson parametric amplifier (JPA) in microwave [26,28], which is the entangled-state signal used in QI protocol, and M independent signal-idler mode pairs are obtained, $\{{\hat a_S^{(k)},\hat a_I^{(k)}} \};1 \le k \le M$. Each T sec-long pulse comprises $M = WT > > 1$ signal-idler mode pairs, where W is the SPDC sources phase-matching bandwidth, as well as the bandwidth of the TMSV signal. Each mode pair is in an identical entangled TMSV with a Fock-basis representation ${|\varphi \rangle _{SI}}$. ${|\varphi \rangle _{SI}}$ is a pure maximally-entangled zero-mean Gaussian state with covariance matrix $V_{SI}^q{\rm{ = }}\left\langle {{{[{{\hat a}_S}{\rm{ }}{{\hat a}_I}{\rm{ }}\hat a_S^\dagger {\rm{ }}\hat a_I^\dagger ]}^T}[\hat a_S^\dagger {\rm{ }}\hat a_I^\dagger {\rm{ }}{{\hat a}_S}{\rm{ }}{{\hat a}_I}]} \right\rangle$, given by

The annihilation operator received in the detector is ${\hat a_R} = \sqrt \kappa {\hat a_S} + \sqrt {1 - \kappa } {\hat a_B}$. There are two cases in radar detection. Case 0 represents no target in the detection area, i.e., $\kappa = 0$; and Case 1 means target presence, $0 < \kappa < 1$, where $\kappa$ represents the total reflectivity in the detection link, ${\hat a_B}$ is in a thermal state with mean photon number ${N_B}/(1 - \kappa )$. Then, we can obtain the covariance matrix for the received signal $V_{RI}^q{\rm{ = }}\left\langle {{{[{{\hat a}_R}{\rm{ }}{{\hat a}_I}{\rm{ }}\hat a_R^\dagger {\rm{ }}\hat a_I^\dagger ]}^T}[\hat a_R^\dagger {\rm{ }}\hat a_I^\dagger {\rm{ }}{{\hat a}_R}{\rm{ }}{{\hat a}_I}]} \right\rangle$, given by

Note that, coherent-state is a concept in quantum optics, and there is no counterpart in the microwave band. In practical radar, we generally use the covariance matrix $E(x{x^T})$ of signal as a tool to analyze the correlation between the probe (signal) and the reference (idler, the classical copy of the signal) [29], where $x = {[{I_1}(t),{Q_1}(t),{I_2}(t),{Q_2}(t)]^T}$ represent the in-phase and quadrature components of the signal and idler. When using the classical state most analogous to the mode pairs created by SPDC, the maximum sub-diagonal element of the covariance matrix $V_{SI}^c$ for classical signal is limited by the standard quantum limit, which is ${N_S}$.

3.2 OPA receiver

Now, we first analyze the processing and output of the received quantum modes in the OPA receiver. The scheme of OPA is shown in Fig. 1. Then we can obtain that for TMSV signal. The average number of photons per mode output by OPA $N_{OPA}^q(i)$ is given by

The joint state of the M received modes state ${\rho _c}$ is the M-fold tensor product $\rho _c^{ {\otimes} M}$. So, after the joint quantum measurement for M modes at the receiver, the probability mass function of the total clicks at the output port $n_{OPA}^{}(i)$ under the two cases can be given. The mean and variance of this distribution are $M{N_{OPA}}$ and $M{D_{OPA}}$, where ${D_{OPA}} = {N_{OPA}}({N_{OPA}}{\rm{ + }}1)$ [22]. Therefore, for one pulse or one detection event with M modes, we can get the mean and variance of the number of photons output by OPA for different signals and different cases, given as

 

Fig. 1. The scheme of the OPA receiver [22]

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3.3 PC receiver

Next, we analyze the processing and output of the received quantum modes in the PC receiver. The scheme of PC receiver is shown in Fig. 2. According to the quantum description of the beam splitter, ${\hat a_X} = ({\hat a_C} + {\hat a_I})/\sqrt 2$, ${\hat a_Y} = ({\hat a_C} - {\hat a_I})/\sqrt 2$, where ${\hat a_C} = \sqrt 2 {\hat a_V} + \hat a_R^\dagger$, it can be obtained that

If we define that ${C_q} = \sqrt {{N_S}({N_S} + 1)}$ and ${C_c} = {N_S}$, then we can get $\left\langle {\hat N} \right\rangle = 2\sqrt \kappa {C_{c/q}}$. According to ${\left\langle {\Delta N} \right\rangle ^2} = \left\langle {{{\hat N}^2}} \right\rangle - {\left\langle {\hat N} \right\rangle ^2}$, the variance of the average number of photons per mode can be given by

Therefore, for TMSV/CS with M modes, the mean and variance of the total clicks at the output of PC receiver $n_{OPC}^{c/q}(i)$ are respectively given by $E[{n_{PC}^{c/q}(0)} ]= 0$, $E[{n_{PC}^{c/q}(1)} ]= 2M\sqrt \kappa {C_{c/q}}$, and

 

Fig. 2. The scheme of the PC receiver [22]

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Note that, when a coherent-state transmitter is used, homodyne detection on each received mode yields a Gaussian-distributed random variable, which has a variance $(2{N_B} + 1)/4$ and a mean 0 or $\sqrt {\kappa {N_S}}$ in different cases [22,30].

4. SNR gain of QI radar and classical radar

Considering that the detector receives not only the target reflected signal and background noise but also the idler used to realize joint processing, the effective SNR at the input of the detector is

Then the gain is ${G_c} = SNR_{output}^c - SN{R_{input}}$ for classical signal processing, and ${G_q} = SNR_{output}^q - SN{R_{input}}$ for quantum signal processing.

In the simulations, parameters are set as follows, $\kappa = 0.1$, ${N_B} = 620$ and $M = {10^6}$. The value of parameter ${N_B}$ is obtained from the average number of thermal noise photons at 3 cm wavelength and 298K room temperature. $M{\rm{ = }}{10^6}$ can be considered as a pulse with 1 GHz bandwidth and 1 ms time-width, which is also the sec-long for one detection event. It should be noted that according to the Gaussian QI theory, QI needs to meet the conditions that ${N_S} < < 1$, ${N_B} > > 1$ and $M > > 1$, in order to maximize its advantage over the optimal CI. The latter two conditions have been satisfied in our assumptions. As mentioned before, quantum advantage ${G_{QE}} = {G_q} - G_c^{\max }$ represents the advantage of QI over optimal CI, where $G_c^{\max }$ is the maximum value of signal processing gain that can be obtained by classical signal, i.e., the SNR gain obtained by homodyne receiver for coherent-state in optics.

As shown in Fig. 3, for both PC and OPA receiver based QI radar, ${G_{QE}} \approx 3dB$ when ${N_S} < < 1$. In general, with the increase of ${N_S}$, the quantum advantage of QI radar gradually disappears. Specifically, when ${N_S} > 10$, the SNR gain of QI radar based on PC receiver is almost the same as that of optimal CI radar, which is basically close to the upper limit allowed by classical physics. While, same as the discussion of the quantum advantage of QI radar in terms of QCB and ROC, the OPA receiver can only operate efficiently at extremely low pump power. In addition, different from the PC receiver, it cannot cancel the common-mode excess noise in ${\hat a_X}$ and ${\hat a_Y}$. Therefore, like the previous analysis of QCB of the OPA receiver, the SNR gain obtained by OPA receiver is slightly lower than that of PC receiver. What’s more, when the signal power does not satisfy the extremely weak condition, the performance of QI radar based on OPA receiver is even worse than the optimal CI radar. Similarly, when the transmitted signal is non-entangled but has the maximum classical correlation, the SNR gain obtained by the PC receiver $G_c^{PC}$ is slightly higher than that of the OPA receiver $G_c^{OPA}$ if the signal power is extremely weak (${N_S} < < 1$). In addition, $G_c^{PC}$ can even achieve the SNR gain of the optimal CI radar with the increase of ${N_S}$. However, as mentioned above, the OPA receiver cannot work effectively when ${N_S}$ is larger, so that its performance cannot reach the optimal CI radar. Furthermore, we can clearly observe from Fig. 3 that with the increase of ${N_S}$, both $G_q^{PC}$ and $G_c^{PC}$ tend to approach the SNR gain $G_c^{\max }$ of the optimal CI radar. The reason for this is that, with the increase of signal power, the benefits of quantum correlation and the effects of quantum noise become increasingly insignificant. It also explains why QI radar has quantum advantage only when the transmitted signal is extremely weak from the perspective of SNR gain obtained by signal processing.

 

Fig. 3. The SNR gain of TMSV / CS after detection and processing by OPA / PC receiver, where the SNR gain is obtained by optimal classical detection. (The solid line indicates the SNR gain for TMSV signal detected by PC receiver; the solid line marked with circles represents the SNR gain for TMSV signal detected by OPA receiver; the dashed line indicates the SNR gain for CS signal detected by PC receiver; the dash-dot-line marked with circles represents the SNR gain for CS signal detected by the OPA receiver; the dotted line shows the SNR gain of homodyne (Hom) detection for coherent-state in optics, which represents upper limit of SNR gain that can obtained in classical system.)

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Summarizing the above analysis, we point out from the perspective of the SNR gain obtained by signal processing (but not the error probability exponent [31]) that,if PC or OPA receiver is used as the detector in the radar system, QI also has 3 dB enhancement compared with the optimal CI, and the performance of PC receiver is better than the OPA receiver. We can also expect that if the optimal quantum receiver of QI is implemented, the quantum enhancement of SNR gain may reach 6 dB that is promised by QI theory. However, due to the lack of appropriate optimal quantum receiver scheme, we have not been able to prove this, which can be our work in the future.

In addition, we find that if QI and CI both use the same receiver, or only take the MF based classical radar as the comparison, the enhancement brought by entangled signal (TMSV state) is higher than the quantum advantage ${G_{QE}}$ when ${N_S} < 1$. We define this enhancement as the quantum correlation enhancement caused by the entangled-state signal, and denote it as ${G_{qc}}$. For the same receiver, whether the OPA or PC receiver, ${G_{qc}} = {G_q} - {G_c}$. Unlike homodyne detection, PC receiver only has the ability to output the phase-sensitive correlation, which cannot realize the optimal signal processing of CS when the signal power is extremely low. Thus, only when ${N_S} > 1$, the signal processing gain $G_c^{PC}$ obtained by PC receiver can approach $G_c^{\max }$. But for $G_c^{OPA}$ of OPA receiver, limited by the operating condition that small pump power, it can ever not reach $G_c^{\max }$. Therefore, we can conclude that PC receiver is a detector that is similar in physical fundamental to MF and more suitable for radar detection. It can be deployed in both QI and CI. Moreover, we found that

According to this, it can be shown that compared with the MF based classical radar, the SNR gain provided by QI radar with PC receiver, is much higher than that of CI when ${N_S} < < 1$. For instance, at the condition of ${N_S} = 0.01$, $G_{qc}^{PC} = 20dB$ is much higher than ${G_{QE}}$. As discussed before, although MF is considered as the optimal receiver under white noise in classical radar theory, in fact, it cannot achieve the maximum classical SNR gain $G_c^{\max }$ when the signal power is extremely low. Hence this enhancement which is higher than the quantum advantage is also of practical significance and cannot be ignored.

5. Conclusion

In conclusion, from the perspective of radar signal processing, we find that the advantage of QI radar is due to the quantum correlation in entangled-state that makes an additional contribution to signal processing. We analyze the SNR gain obtained by the realizable OPA / PC receiver after detecting and processing for TMSV and CS signals respectively (corresponding to QI and CI), as well as the classical optimal SNR gain achieved by homodyne detection for coherent-state. We show that QI based on OPA/PC receiver has about 3 dB enhancement in the SNR gain obtained from signal processing compared with the optimal CI, and PC receiver has better performance than OPA receiver. We further find that QI can provide higher gain than the MF based classical radar, which is considered as the optimal radar under white noise, and the gain even can reach tens of dB.

Finally, we must point out that although QI shows its advantages, this advantage is limited to the case of very weak transmitted signal power, so it may be a challenge for applying QI to radar remote detection. However, the research on QI is still in its infancy. We note that some potential solutions have been proposed to solve the above problems. These schemes include multiple input multiple output (MIMO) quantum radar system and noise free amplification technology for quantum signals, such as quantum phase preserving amplifiers [32–34]. Therefore, we should not rush to deny its practicability in radar, but need more in-depth research. Generally speaking, the existence of super-classical advantages brought by entanglement has been proven. Taking advantages of this correlation makes it possible for us to break through the bottlenecks in some classical technologies.