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The Quantum Illumination Story

Superposition and entanglement, the quintessential characteristics of quantum physics, have been shown to provide communication, computation, and sensing capabilities that go beyond what classical physics will permit. It is natural, therefore, to explore their application to radar, despite the fact that decoherence—caused by the loss and noise encountered in radar sensing—destroys these fragile quantum properties. This paper tells the story of “quantum illumination”, an entanglementbased approach to quantum radar, from its inception to its current understanding. Remarkably, despite loss and noise that destroy its initial entanglement, quantum illumination does offer a target-detection performance improvement over a classical radar of the same transmitted energy. A realistic assessment of that improvement’s utility, however, shows that its value is severely limited. Nevertheless, the fact that entanglement can be of value on an entanglement-breaking channel—the meta-lesson of the quantum illumination story—should spur continued research on quantum radar.

Superposition and entanglement-Schrödinger's cat being simultaneously alive and dead [1], and the "spooky action at a distance" that Einstein found disturbing [2]-are quantum-mechanical phenomena that are moving from fundamental studies into scientific and engineering applications. Quantum computers, if realized at sufficiently large scale, will vastly outstrip the capability of classical machines for a variety of problems in simulation [3], optimization [4], and machine learning [5]. Those large-scale quantum computers-running Shor's quantum factoring algorithm [6]-will also break the public-key infrastructure on which Internet commerce currently relies. Quantum communication, in the form of quantum key distribution [7], however, may thwart that quantum threat. In other work, quantum-enhanced sensing is moving out of the laboratory and into real use, the most notable example being the incorporation of squeezed-state light into the laser interferometric gravitational-wave observatory (LIGO) [8]. A natural question to ask, therefore, is whether quantum techniques can bring performance gains to radar sensing. This paper will follow one avenue of quantum radar research from its inception to now: quantum illumination.

I. LLOYD'S QUANTUM ILLUMINATION
Lloyd [9] coined the term "quantum illumination" for his entanglement-based approach to improving an optical radar's capability to detect a weakly-reflecting target embedded in background noise that can be much stronger than the target return. His work, which built on Sacchi's earlier studies of quantum operation discrimination [10], [11], compared the targetdetection performance for the two scenarios shown in Fig. 1. In both scenarios an optical transmitter illuminates a region of space in which a weakly-reflecting target is equally likely to be absent or present within always-present background light. In Fig. 1(a), the transmitter's signal beam is a sequence of N high time-bandwidth product (M = T W 1), single-photon pulses. The receiver, for this single-photon (SP) scenario, makes a minimum error-probability decision between hypotheses H 0 (target absent) and H 1 (target present) from observation of the light returned from the interrogated region. In Fig. 1(b), the transmitter illuminates the region of interest with a sequence of N high time-bandwidth product (M = T W 1), single-photon signal pulses, each of which is entangled with a companion single-photon idler pulse; see Appendix A for the details. The receiver, for this quantum illumination (QI) scenario, makes its minimum error-probability decision between H 0 and H 1 from observation of the retained idler light and the light returned from the interrogated region.
? ? The crucial assumptions in Lloyd's analysis of the unentangled and entangled scenarios are as follows.
• When the target is present, the roundtrip transmitter-to-target-to-receiver transmissivity for the signal beam is 0 < κ 1.
• The background light's average photon number per temporal mode, N B , satisfies the low-brightness condition N B 1.
• For each transmitted signal pulse, at most one photon is returned to the receiver, regardless of whether the target is absent or present, implying that M N B 1.
Under these assumptions, Lloyd identified two operating regimes, the "good" and the "bad", for his SP and QI scenarios, and compared their error probabilities' quantum Chernoff bounds [12] in these regimes. In their good regimes, SP and QI's error probabilities have the same bound, Pr(e) SP ≤ e −N κ /2 and Pr(e) QI ≤ e −N κ /2, which, because N κ equals the average number of signal photons returned when the target is present, whereas 0 is the average number of signal photons returned when the object is absent, equals the signal shot-noise limit for laser communication with on-off-keying (OOK) modulation. Despite (1)'s applying to SP and QI's good regimes, QI still enjoys a substantial good-regime performance advantage over SP, because SP's good regime is limited to κ N B , whereas QI's extends to the much larger parameter region in which κ N B /M .
The comparison between SP and QI operation is radically different in their bad regimes.
Here Lloyd found that and These bounds mimic the background-limited error probability of OOK laser communication, with N B being the background's brightness for SP operation, and N B /M being that brightness for QI operation. The bad-regime results reveal a double advantage for QI: first, its bad regime applies in a smaller κ range than that of SP; and second, when both systems are in their bad regimes, QI enjoys a factor-of-M greater error-probability exponent (effective signal-to-noise ratio). At optical frequencies, high time-bandwidth product is easy to obtain, e.g., a 1 µs pulse at 300 THz center frequency (1 µm wavelength) with 1 THz (1/3-percent fractional) bandwidth yields M = 10 6 , in which case Lloyd's bad-regime QI has a 60 dB higher effective signal-to-noise ratio than its SP competitor. Moreover, and remarkably, this advantage is afforded despite the background noise's destroying the initial entanglement, i.e., the retained and returned light are not entangled.
Despite the idealized nature of Lloyd's QI analysis-it presumes an on-demand source of high-T W entangled signal-idler photon pairs, lossless idler storage, and perfect realization of an optimum quantum receiver-its enormous predicted performance enhancement in the bad regime motivated a great deal of follow-on research. Some of that research, unfortunately, took much of the air out of QI's balloon, as we will describe below. Before doing so, however, a brief preface about classical versus quantum radar is in order.
We were careful, earlier in this section, not to describe the comparison between SP and QI operation as one between a classical radar (SP) and a quantum radar (QI). In quantum optics, see, e.g., [15], [16], it is conventional to reserve the appellation "quantum" for those systems whose performance analysis requires the quantum theory of photodetection. In particular, their performance cannot be correctly quantified from the semiclassical theory of photodetection, in which light is treated as a classical (possibly stochastic) electromagnetic wave and the discreteness (quantization) of the electron charge gives rise to shot noise. Furthermore, measurements of light beams that are in coherent states [17], or classically-random mixtures thereof, in any of the three basic photodetection paradigms-direct detection, homodyne detection, or heterodyne detection-do not require quantum photodetection theory to obtain correct measurement statistics [15]. Hence such states are called classical states.
Systems that employ nonclassical states-states other than coherent states or their random mixtures-require the use of quantum photodetection theory. In this regard, we note that single-photon states and entangled states are not classical states, and hence Lloyd's performance comparison is between two quantum radars, one of which employs entanglement (QI) while the other (SP) does not.
A coherent state-other than the zero-photon (vacuum) state-contains a random number of photons. So, Shapiro and Lloyd [18] compared Lloyd's QI to a classical radar by replacing Lloyd's SP transmitter in Fig. 1(a) with a coherent-state transmitter (an ideal laser) that produces a sequence of N pulses, each of which has unity average photon number. The quantum Chernoff bound for their coherent-state radar, which applies for all 0 ≤ κ ≤ 1 and for all N B ≥ 0, reduces to   Pirandola and Lloyd [20] and where, in both cases, 0 < κ 1, N S 1, and N B 1 are assumed. By analogy with Lloyd's work, we might term this operating regime the "bad" regime for Tan   on Pr(e) CS from Ref. [19]. It is important to remember that quantum Chernoff bounds are known to be exponentially-tight upper bounds, while it is known that the Bhattacharyya lower bound is always loose. Furthermore, Tan et al. showed that the CS radar affords the lowest error probability of any classical-state radar of the same average transmitted energy.
Thus, because Fig. 3 shows Pr(e) UB QI becoming lower than Pr(e) LB CS for sufficiently high M values, it provides definitive proof that a QI radar can, in principle, outperform all classical radars of the same transmitted energy for the target-detection scenario addressed by Tan et al. We will say more later about why we refer to Fig. 3's QI advantage over CS operation as being "in principle". For now, it suffices to point out that when Ref. [19] appeared there was no known receiver that provided any QI performance advantage over CS operation. Why that was so and how it was overcome require some understanding of quantum photodetection theory and the quantized electromagnetic field's Gaussian states. So those topics are next on our agenda.
The mth temporal modes of Tan et al.'s QI signal and idler from (B1) and (B2) have associated photon annihilation operators,â Sm andâ Im , whose adjoints are the photon creation operators,â † Sm andâ † Im . These names originate from the operators' actions on their andâ † Km |n Km = √ n + 1 |n + 1 Km , for n = 0, 1, 2, . . . , and K = S, I, where the ket vectors {|n Km : n = 0, 1, 2, . . . , } represent states containing exactly n photons. Ideal photon counting on the mth signal and idler modes measures the photon-number operatorsN Sm ≡â † Smâ Sm andN Im ≡â † Imâ Im , respectively. It then follows that ideal photon counting on the coherent state given in (B4) yields a Poisson-distributed output with mean N S , as expected from the coherent state's photodetection statistics being obtainable from semiclassical (shot-noise) theory [15].
The signal and idler's mth temporal modes have operator-valued quadrature components, for K = S, I, and ideal (quantum-limited) optical homodyne detection with the appropriate local oscillator fields measures these operators [15]. The positive-operator-valued measurements (POVMs) associated withâ Sm andâ Im [21] can be realized by ideal (quantumlimited) optical heterodyne detection [15]. Although heterodyne detection provides informa-tion about both quadratures, the Heisenberg uncertainty principle forces this measurement to incur extra noise on each quadrature that is not present in homodyne measurement of a single quadrature [15].
Gaussian states of the mth signal and idler modes,â Sm andâ Im , are the quantum analogs of classical, complex-valued Gaussian random variables, a Sm and a Im . Thus, Gaussian states of these two modes are completely characterized by knowledge of their first and second moments [21], â Km , ∆â † Km ∆â Jm , and ∆â Km ∆â Jm , where K = S, I, J = S, I, · denotes ensemble average, and ∆â Km ≡â Km − â Km . The coherent state from (B4) is a Gaussian state with â Sm = √ N S , and ∆â † Sm ∆â Sm = ∆â 2 Sm = 0. The entangled signal-idler state from (B3) is also Gaussian. It is a two-mode squeezed vacuum (TMSV) state [21], whose measurement statistics are completely characterized by â Sm = â Im = 0, â † Smâ Sm = â † Imâ Im = N S , â 2 Sm = â 2 Im = 0, â † Smâ Im = 0, and â SmâIm = N S (N S + 1). The preceding brief introduction to Gaussian states provides enough information to understand the origin of Tan et al.'s QI advantage and why conventional optical receiversdirect detection, homodyne detection, and heterodyne detection-do not realize any of that advantage. The cross correlations of all zero-mean classical signal-idler states must obey [22] and For arbitrary, zero-mean quantum states (11) applies, but (12) becomes the less restrictive The TMSV is a zero-mean Gaussian state that violates (12), because | â SmâIm | = Thus it is a nonclassical state, as we already knew from (B3), and in fact maximally entangled, because it saturates the bound in (13). Furthermore, withâ Rm = √ κâ Sm + √ 1 − κâ Bm being the photon annihilation operator for the returned light's mth mode when the target is present, whereâ Bm is the photon annihilation operator of the relevant background-light mode [19], Tan et al.'s QI system has the conditional cross-correlation given the target-present hypothesis H 1 . When N S 1, this conditional cross-correlation, which is the signature of target presence in Tan et al.'s QI system because â RmâIm H 0 = 0, greatly exceeds the √ κ N S classical-state limit on this cross correlation that applies when â † Smâ Sm = â † Imâ Im = N S . When N S 1, however, QI's target-presence signature is only slightly better than the classical limit [24]. So, it should be no surprise that the preferred operating regime for Tan et al.'s QI system has low signal brightness, N S 1. That this preferred operating regime has high background brightness, N B 1, follows from Nair's no-go theorem [25], which shows that QI offers only an inconsequential performance improvement over CS operation's Pr(e) CS ≤ e −M κN S /2 when background light can be neglected.
It might seem it would be easy to build a receiver capable of reaping the benefit associated with QI's enhanced cross-correlation signature in the 0 < κ After all,â RmâIm can be expanded into four terms that are products of the return and idler modes' quadrature components, and these quadrature components can be measured individually by means of homodyne detection. Unfortunately, this approach does not work, because we need to measure both quadratures of the return and idler modes, and the Heisenberg uncertainty principle precludes that being done, e.g., by heterodyne detection, without incurring additional noise. The net effect is that neither homodyne detection nor heterodyne detection can be used to provide any QI target-detection performance advantage over its CS counterpart. Direct detection (time-resolved photon counting) is also unable to provide a QI advantage. Although SPDC produces signal and idler photons in time-coincident photon pairs, photon-coincidence counting-as routinely used, e.g., in SPDC ghost imaging [16]-does not provide a usable QI signature for target presence. This failure is because the presence of high-brightness background light in the return implies that every detection of an idler photon will have a coincident detection from the returned light, regardless of target absence or presence.
The signature of target presence in Tan et al.'s QI system is the conditional phase-sensitive cross correlation between its returned light and its retained idler, namely â RmâIm H 1 for those beams' mth modes. The returned and retained light's target-present phase-insensitive cross correlation, â † Rmâ Im H 1 , can be measured in second-order interference between the two beams. But Tan et al.'s QI system has â † Rmâ Im H j = 0 for j = 0, 1, i.e., regardless of whether the target is absent or present, and its â RmâIm H j cannot be measured in second-order interference; see Appendix C for the details. Guha and Erkmen [26] recognized this problem, and offered a partial solution with their optical parametric amplifier (OPA) receiver. That receiver, its experimental realization, and other aspects of obtaining QI's performance advantage are treated in the next section.

III. QI RECEIVERS AND EXPERIMENTS
To understand Guha and Erkmen's OPA receiver requires some results from the quantum theory of nonlinear optics. Because that theory also underlies the SPDC behavior we have already been employing, a brief introduction encompassing both SPDC and OPA operation is germane.
In crystals that have a second-order nonlinear susceptibility, such as lithium niobate (LiNbO 3 ) or potassium titanyl phosphate (KTiOPO 4 ), a strong pump beam at frequency ω P can interact with weak signal and idler beams at lower frequencies ω S and ω I satisfying ω S + ω I = ω P . Continuous-wave SPDC has no inputs at the signal and idler frequencies.
Nevertheless, it produces outputs at those frequencies. These outputs can be regarded as arising from a photon-fission process, in which a single pump photon splits into a signal-idler photon pair. Energy conservation at the single-photon level requires thathω S +hω I =hω P .
Momentum conservation at the single-photon level requires thathk S +hk I =hk P , where k J , for J = S, I, P , are the propagation vectors for the signal, idler, and pump photons.
Even with these conservation conditions obeyed-which can only be achieved within the crystal's phase-matching bandwidth, |ω K − ω Ko | ≤ πW for K = S, I, about the signal and idler's center frequencies, ω So and ω Io -continuous-wave SPDC produces only pW of signal and idler per mW of pump power, see, e.g., [27].
For the QI application, we want the signal and idler to be single-spatial-mode fields, so the former can be formed into a tight transmitter beam, which is why our description of Tan et al.'s QI system-and, similarly, of Lloyd's QI system-only considered temporal modes. The Gaussian-state treatment of single-spatial-mode SPDC [28] shows that the signal's frequency Guha and Erkmen's receiver for Tan et al.'s QI system uses a continuous-wave OPA operating at extremely low gain, which means that it can be realized with a crystal identical to the one used for that system's SPDC source. In the receiver, however, the returned light and the retained idler light are applied as the signal and idler inputs to the OPA crystal, and the idler output's mth mode is given byâ out where the gain, G > 1, is chosen to optimize target-detection performance. Ideal photon counting is done on all idler-output modes, which measures their total photon-number operator, Although written as a sum of modal photon-number operators, theN T measurement is easily realized by photon counting on the OPA's duration-T idler output in response to the duration-T returned light and retained idler inputs. The conditional means-given target absence or presence-for theN T measurement are as follows: Here in QI's usual 0 < κ 1, N S 1, N B 1, operating regime, implying a 3 dB improvement in error-probability exponent over Tan et al.'s CS system of the same transmitted energy.
Implicit in (17) are two important assumptions made by Guha and Erkmen, both of which arise from the interferometric nature of applying the returned light and the retained idler to the signal and idler inputs of a low-gain OPA. The first assumption is that the idler is stored losslessly. The second is that the idler storage is matched in time delay and phase to those of the light returned from the target (when it is present). We will revisit these assumptions later in discussing QI's utility for realistic radar scenarios. For now, it suffices to note that Zhang et al. [29] reported the first experimental demonstration of Tan where |ω So + ω S (|ω Io − ω I ) denotes a single-photon signal (idler) of frequency ω So + ω (ω Io − ω) and, as has been implicitly assumed earlier, the signal (idler) brightness is taken to be constant over the phase-matching bandwidth. If this photon pair illuminates a crystal identical to the one used for its SPDC generation, then, with extremely low probability, sumfrequency generation (SFG) can occur, viz., a photon-fusion process in which the signal-idler photon pair is converted to a single photon at the pump frequency, ω P = ω So +ω Io . Zhuang et al. realized that SFG's being a coherent process involving all of QI's mode pairs offered a path to QI reception that was not bound by the limitations of LOCC operation. Nevertheless, their work made the rather significant assumption that SFG could be done with 100% efficiency at the photon-pair level, something that is far beyond the current capability of nonlinear optics. Furthermore, the presence of high-brightness background light drove them to using multiple cycles of SFG and photon-counting measurements, but ideal realization of these steps resulted in a receiver whose quantum Chernoff bound for QI matched that found by Tan  determination of QI's receiver operating characteristic (ROC), i.e., its detection probability, P D , versus false-alarm probability, P F , when optimum quantum reception is employed [37].
QI's ROC is crucial because it is a far better target-detection performance metric than error probability, as radar targets should not be presumed equally likely to be absent or present.
Not surprisingly, QI's ROC improvement over CS operation of the same transmitted energy turned out to be equivalent to a 6 dB increase in effective SNR [37].
QI's aforementioned performance advantages-in error probability or ROC-assume that the target return, when present, has known amplitude and phase, a situation that seldom occurs in light detection and ranging (lidar) applications. At lidar wavelengths, most target surfaces are sufficiently rough that their returns are speckled, i.e., they have Rayleighdistributed amplitudes and uniformly-distributed phases. QI's OPA receiver-which affords a 3 dB-better-than-classical error-probability exponent for a return with known amplitude and phase-fails to offer any performance gain for Rayleigh-fading targets. The sumfrequency generation receiver from Ref. [35]-whose error-probability exponent for a nonfading target achieves QI's full 6 dB advantage over optimum classical operation-outperforms the classical system for Rayleigh-fading targets [38]. In this case, however, QI's advantage is subexponential under ideal operating conditions, so that its benefit is far more vulnerable to nonidealities such as were encountered in Zhang et al.'s OPA receiver for the non-fading scenario.

IV. MICROWAVE QUANTUM ILLUMINATION: CONCEPT AND REALITIES
Lloyd's QI presumed operation at optical wavelengths, because high-sensitivity photodetection systems have long been limited by noise of quantum-mechanical origin. Hence the optical region was the natural setting in which to seek a quantum advantage. Tan and nighttime N λ (and hence N B ) values are several orders of magnitude lower. So, the advantage expected from ideal QI operation at optical frequencies would only accrue were there bright-light jamming. Consequently, a significant amount of interest in QI from the radar community only arose after Barzanjeh et al. [41] proposed an approach for doing QI in the microwave region, where the naturally-occurring background does satisfy N B 1 [42], and most target-detection radars operate.  [26] had previously shown that a PC receiver's performance advantage over a conventional radar is equivalent to that of an OPA receiver, i.e., a 3 dB advantage in error-probability exponent under ideal conditions. Even though this advantage could easily fall prey to system nonidealities, the fact that QI was now predicted to offer an advantage at microwave frequencies ignited a great deal of attention from the radar community, prompted in part by some inaccurate reporting [43]. It is now time, therefore, to turn our attention to those nonidealities, some of which were discussed in Refs. [41], [44], and confront the realities of seeking a QI advantage for target detection.
We will begin that assessment with the issue of idler-storage loss.
in QI's preferred 0 < κ 1, N S 1, N B 1 operating regime. Comparing these results with (6)'s quantum Chernoff bound for a CS radar shows that 6 dB of idler loss will eliminate QI's performance advantage for FF-SFG or SFG reception, and 3 dB of idler loss suffices for that purpose for OPA or PC reception. Barzanjeh et al. [41] was the first QI target-detection paper to explicitly comment on the adverse impact of idler-storage loss, noting that the idlerstorage loss incurred in the best presently-available technique for optical idler storage-an optical-fiber delay line-precludes their system's offering any performance advantage for targets more than 11.25 km away.
Even were the preceding gloomy assessment of long-range, microwave-QI target detection obviated by a breakthrough in idler-storage technology, microwave QI would still suffer from its difficulty in obtaining a sufficiently high time-bandwidth product with pulse durations that make sense for microwave radars. High time-bandwidth product is critical for QI because-unlike a conventional radar, whose performance improves at constant pulse duration and bandwidth with increasing transmitter power in the absence of clutter-QI systems' performance advantage degrades at constant pulse duration and bandwidth with increasing transmitter power [44]. So, even though Barzanjeh et al.'s EOM converters-which are intrinsically narrowband-could be replaced with broadband microwave-entanglement generation in a traveling-wave parametric amplifier [45], the available time-bandwidth products in the microwave region are still dwarfed by what is easily achieved at optical wavelengths.
For example, the 1/3-percent fractional bandwidth at 1 µm wavelength that gives a 1 µs pulse duration a 10 6 time-bandwidth product only gives that pulse duration a 10 2 time-bandwidth product at 1 cm wavelength. Pushing to mm-wave operation with higher fractional bandwidth and longer pulse duration will afford considerably higher time-bandwidth product, but that will run afoul of another consideration: QI's single-bin-per-pulse interrogation limitation [44].
Unlike a conventional radar, QI can only interrogate a single polarization-azimuthelevation-range-Doppler resolution bin at a time [46], if it is derive its full performance advantage over a conventional radar. Interrogation of K B bins simultaneously, by splitting the stored idler into K B equal-strength pieces to make said measurements, leads to a 10 log 10 (K B ) dB performance loss for each bin. So, assuming ideal equipment, simultaneous interrogation of two resolution bins sacrifices the entire performance advantage of an OPA receiver, and simultaneous interrogation of four resolution bins gives up the entire performance advantage of an FF-SFG receiver. In this regard it is important to note that optical amplification cannot be used, prior to idler splitting, to mitigate the preceding problem.
This failure is because the amplifier's unavoidable amplified spontaneous emission noise will be much stronger than the amplified idler, making the latter useless for QI. There is another resolution-related problem with QI: the target should lie entirely within a single polarization-azimuth-elevation-range-Doppler resolution bin throughout the signal pulse's full duration. If such is not the case, there will be a mismatch between the temporal behavior of the returned radiation's target-present component and the temporal behavior of the stored idler. This mismatch, which will degrade the performance of the QI receivers we have considered, is quantified by a normalized overlap integral, 0 < κ m ≤ 1, that reduces QI's error-probability exponent by a factor of κ m . The mismatch problem is especially significant for optical QI, where a 1 THz bandwidth implies that <1 mm target range extent is needed to ensure κ m ≈ 1.
Some additional points worth noting are as follows: (1)  made with both a classically-correlated-noise radar and a coherent-state radar with incoherent post-heterodyne processing. As the next section will show, their reception techniques preclude their QI experiments from outperforming an optimized classically-correlated-noise (CCN) radar. For that reason, we will refer to their QI setups as quantum-correlated-noise (QCN) radars, as opposed to a true QI radar, viz., the Tan et al. scenario with an OPA receiver, which outperforms all classical radars of the same transmitted energy [52].

V. CORRELATED-NOISE RADARS: QUANTUM VERSUS CLASSICAL
Here, for greater generality in our consideraton of correlated-noise radars, we will usê to model the returned light's M modal annihilation operators when the target is present, where 0 < κ 1 and 0 ≤ θ ≤ 2π may be deterministic and known, as in Tan et al. [19] (κ 1 known and θ = 0), or random with a known joint probability distribution, as in Zhuang et al. [37] ( √ κ and θ statistically independent, with a Rayleigh-distributed √ κ having κ 1, and a uniformly-distributed θ). We will also assume that these noise radars use heterodyne (or preamplified-heterodyne) detection, with noise figures N F ≥ 1, where 1 is the ideal (quantum-limited) value [53].
The QCN radar we shall consider uses an SPDC source and heterodyne (or pre-amplified heterodyne) detection at the signal and idler frequencies. and where N R ≡ κN S + N B and C q ≡ κN S (N S + 1).
The CCN radar we will consider uses a high-brightness classical noise source whose output is divided-with an appropriate splitter-into a low-brightness signal and a high-brightness idler. Like the QCN radar, the CCN radar will employ heterodyne (or pre-amplified heterodyne) detection of its retained idler and its return from the region interrogated by its signal.
Conditioned on the true hypothesis and the κ, θ values, the CCN radar's M mode-pair outputs are also a set of independent, identically-distributed, complex-valued, 2D random column vectors whose quadrature components have zero-mean Gaussian distributions, but now with covariance matrices and where C c ≡ √ κN S N I .
Note that SPDC forces the signal and idler brightnesses at the QCN radar's transmitter to be identical, whereas, by starting with a high-brightness classical noise source and using a highly-asymmetric splitting ratio, the CCN radar's transmitter can have a low-brightness (N S 1) signal that matches that of the QCN radar while retaining a high-brightness (N I 1) idler. As we will soon see, operating the CCN radar with N I 1 N S is a crucial point that was missed in the microwave QI experiments to date [49]- [51]. and Similarly, given the true hypothesis and the κ, θ values, the CCN radar's transformed mode pairs are independent and identically distributed with zero-mean Gaussian-distributed quadrature components having covariance matrices and It is now clear that the QCN and CCN radars have identical conditional statistics in the limit N I → ∞ when their detectors are quantum limited (N F = 1). Hence we expect that their quantum-limited performance will be virtually the same for N I 1. Moreover, for non-ideal detectors (N F > 1), the CCN radar outperforms the QCN radar when N I > N F (N S + 1)/(N F − 1), because the CCN radar's transformed idler modes then have lower noise than those of the QCN radar, while all the other second moments of the two radars are identical. Furthermore, these conclusions apply for κ, θ deterministic and known, as well as for κ, θ random with a known joint probability distribution. Thus we have proven that the QCN radar cannot outperform all classical radars of the same transmitted energy and identical detector capabilities.

VI. ROC COMPARISONS
As a capstone for all that has been presented, this section compares the ROCs for five radars in the best possible operating scenario: quantum-limited detection, and κ, θ deter-  Figure 5 shows the ROCs for these radars, plotted as log 10 (P M ) versus log 10 (P F ) for P F , P M ≤ 0.5. The QCN, CCN, and QI-OPA ROCs were computed with Van Trees's ROC approximation technique [54], which is known to be accurate for M 1 and especially so [55] for Gaussian signals in Gaussian noise, as we have for the QCN and CCN radars.
The CS-Het ROC [54], where with I 0 (x) being the zeroth-order modified Bessel function of the first kind, was computed exactly, as was the CS-Hom ROC [54], where with Q −1 (·) being its inverse function, which obeys Q −1 [Q(x)] = x for −∞ < x < ∞. require phase information: the CS-Hom radar needs that information to lock its local oscillator's phase to that of the expected target return, while the QI-OPA radar needs it to lock its retained idler's phase to that of the expected target return. One final comment about QCN versus CCN radars is in order: the QCN radar requires a dilution refrigerator to house its Josephson junction parametric amplifier source, but the CCN radar might use a room-temperature microwave noise generator for its source.

VII. CONCLUDING REMARKS
In conclusion, despite QI target detection's not being an immediate boon to the radar community, it is important to remember that it is the first example of a bosonic system that offers a performance improvement on an entanglement-breaking channel when its transmitter is subject to an energy constraint [56]. In other words, we should not dismiss the value of entanglement for the lossy, noisy situations that are the norm in microwave radar. Indeed, this meta-lesson may lead to other quantum systems that offer real utility. Two such examples have already arisen in secure communication: (1) floodlight quantum key distribution (FL-QKD) [59], [60], which directly descends from a communication version [61] of QI target detection, is capable of Gbps secret-key rates over metropolitan-area fiber connections; and The CCN radar's source produces M independent, identically-distributed, signal-idler mode pairs, {(â Sm ,â Im )}, that are in classically-random mixtures of coherent states whose eigenvalues, {(a Sm , a Im )}, have the joint probability density, where δ(·) is the unit-impulse function. In the limit N I → ∞, quantum-limited heterodyne detection of the {â Im } modes yields a noise-free measurement of the {α Im = N I /N S α Sm }.
Optimum post-heterodyne processing is then a maximal-ratio combiner [54], which yields the conditional ROC, given the {α Sm }. In the limit M → ∞ we have that which matches the CS-Hom radar's ROC when N B 1. Note that the preceding nearequivalence also applies to CCN and CS-Hom radars with N F > 1 noise figures that satisfy