Gaussian State-Based Quantum Illumination with Simple Photodetection

Proofs of the quantum advantage available in imaging or detecting objects under quantum illumination can rely on optimal measurements without specifying what they are. We use the continuous-variable Gaussian quantum information formalism to show that quantum illumination is better for object detection compared with coherent states of the same mean photon number, even for simple direct photodetection. The advantage persists if signal energy and object reflectivity are low and background thermal noise is high. The advantage is even greater if we match signal beam detection probabilities rather than mean photon number. We perform all calculations with thermal states, even for non-Gaussian conditioned states with negative Wigner functions. We simulate repeated detection using a Monte Carlo process that clearly shows the advantages obtainable.


I. INTRODUCTION
Quantum states of light for object detection in a noisy environment, coined as "quantum illumination", were originally introduced [1] and subsequently investigated for continuous variable Gaussian states [2][3][4]. Quantum illumination uses quantum correlations to provide improved object detection when compared to classical light sources. The proof of this advantage, at optical frequencies or in quantum radar signal discrimination, boils down to an optimization problem focused on minimizing the probability of error in hypothesis testing; from thereon parameters are chosen to present a favourable picture where quantum states gain a performance advantage over illumination with coherent states. The advantage of entangled Gaussian states prevails in the lossy and noisy scenarios if both modes can be measured "optimally" [5] even if the entanglement is completely lost [6,7]. Then quantum correlations [8] may remain in the form of quantum discord [9].
The problem can be stated in terms of quantum optical state discrimination, which is a well-studied subtopic in quantum information [10]. The goal is to detect successfully an object of weak reflectivity κ possibly hidden in a strong thermal background noise. In effect we wish to determine which of two possible hypotheses, H 0 -the object is absent, or H 1 -the object is present, is true. If we send signal state with density matrixρ for target detection then eitherρ 0 (corresponding to H 0 being true) orρ 1 corresponding to H 1 ) will return, conditioned by the absence or presence of the object. How distinguishable are these two states? Could we measure this difference? Previous analysis has circumvented what measurement we have to make and successive proofs are based on the Helstrom bound, which states that the absolute error in the optimal measurement is proportional to 1 − 1 2 |ρ 0 −ρ 1 | [11]. This is usually difficult to evaluate for Gaussian states, especially for states corresponding to multiple observation trials. Hitherto, a workaround solution has been to use easier-to-calculate fidelity bounds such as Uhlmann fidelity or the Quantum Chernoff bound [12][13][14][15] to estimate the error probability of the optimal measurement [16]. However, this does not resolve the issue of identifying the exact measurement protocol required to perform discrimination. Nevertheless, quantum illumination has been used and measured experimentally at optical frequencies with CCDs [17], sum-frequency generating receivers [18], by fibre-coupled avalanche photodiodes [19], and more recently in the microwave domain [20][21][22][23][24].
In this paper we analyse an experimentally simpler approach to previous state discrim-ination methods for quantum illumination: a direct measurement strategy using positive operator-valued measures (POVMs) that model Geiger-mode photodetectors which can either fire or do not fire when EM radiation falls upon them or not respectively -"click" or "no-click" (see Fig. 1 for difference in schematic set-up). Our approach uses entirely continuous-variable quantum information [25][26][27] with Gaussian states that are fully characterised by their statistical moments. In this way, some computational hurdles of discrete Fock-basis representations (large Hilbert spaces for multi-mode or higher energy states) are avoided by working in the Gaussian-only domain. An essential point of the analysis is that it allows us to perform all calculations exactly within this Gaussian quantum optical framework, even though the states are not all Gaussian. Moreover, since this is a statistical analysis of QI, it is frequency independent and could therefore be applied both to lidar and to radar. By employing this simple click detection strategy, we can easily calculate the click probability of the return signal after its interaction with an external object, whilst incorporating any associated quantum efficiencies and thermal background noise sources, including detector dark counts. We compare results from two signal states with different photon-statistics: a coherent state that has classical photon-statistics and a quantum state heralded via click detection from the entangled two-mode squeezed vacuum (TMSV) [28].
As click-detection is not phase-resolving, the moments of the signal state, even after conditioning, can be parameterised by its average photon numbern = â †â , aiding further simplification. Our purpose is to highlight where quantum illumination may provide an advantage over classical. Classical radar and lidar [29] are well developed technologies with many applications so quantum illumination will not be better in every situation, although some of its advantages for rangefinding have been pointed out recently [30]. We emphasise that there are states and measurement strategies that could give better results for object detection than we obtain, but these would typically require a local oscillator that is phaselocked to the signal. If the distance to the object is uncertain by more than a small fraction of a wavelength they are rendered impotent.
The paper is structured as follows. We give in Section II a brief introduction to continuous-variable bosonic Gaussian states and include some useful identities, before in Sections III and IV introducing direct detection and the conditional states produced from performing a click-detection on one mode of the TMSV in a heralding process. In Section V we incorporate this into quantum illumination theory and compare results from coherent states and TMSV states. Specifically we calculate conditional probabilities of click detection for classical or quantum illumination in the presence or absence of a target object. We use these probabilities to calculate retrodictive conditional probabilities [31] of the presence or absence of an object given a click or otherwise at a detector. In Section VI we describe a simple count probability-matching based means of increasing the advantage obtained by quantum illumination. Section VII describes the simulation of the repeated application of the process described in earlier sections to give the multi-shot scenario. Monte Carlo simulations clearly show the available advantages of quantum illumination with click detection heralding. Finally we add some concluding remarks in Section VIII.

II. BOSONIC GAUSSIAN STATES
For n quantum harmonic oscillator modes, in natural units with = 1, a global quadrature operator can be written in the following alternating order where Ω is the global symplectic identity (symplectic form), that shows the basic phase space  [32], that is, the average of the displacement operatorD(Λ), with Λ = 1 √ 2 {Re(ξ 1 ), Im(ξ 1 ), . . . , Re(ξ n ), Im(ξ n )} the vector of real-numbers corresponding to complex global displacement ξ = {ξ 1 , ξ * 1 , . . . , ξ n , ξ * n }. If the Wigner function is a Gaussian distribution, then the characteristic function must also be Gaussian. Corresponding states are therefore called Gaussian statesρ G and they are completely characterised by the first and second statistical moments of the Wigner functions.
These are expressed by the mean µ := x and the covariance matrix Σ, which has entries For a single mode Gaussian state, n = 1, in this phase-space picture, µ denotes displacement of the state from the vacuum (phase space origin) and the covariance matrix describes the widths of the distribution, the uncertainty surrounding µ. All proper Gaussian states exhibit quadrature noise when viewed in the Wigner function representation. In relation to physical parameters, the isotropic spread of the covariance indicates presence of thermal noise (a circular 2D Gaussian). Unequal distribution of the Gaussian widths along perpendicular axes indicate noise squeezing (elliptical 2D Gaussian), for which the quadrature uncertainties must obey Heisenberg's uncertainty relation ∆ 2 q ∆ 2 p ≥ 1/4.
Conveniently the Gaussian state's Wigner function exists in non-integral form, as a multivariate Gaussian State overlaps become convolution integrals in phase-space Evolution of the quadrature operators via Gaussian unitary operations, that is, unitaries with creation and annihilation operators up to the second order in the interaction Hamiltonian, evolve viax k →Ûx kÛ † . These interactions become symplectic transforms on µ and Σ where S is the symplectic matrix (a real 2n × 2n matrix satisfying SΩS t = Ω and det S = 1) that representsÛ in phase space. The vector µ D is a translation caused by the displacement operation. The covariance Σ is symplectically diagonalized by casting it in the form SΣ ⊕ S t such that Σ ⊕ is a diagonal matrix with repeated eigenvalues ν k =n k + 1 2 ,n k = â † kâ k , that shows the global Gaussian state comprising of irreducible thermal states. Tensor products of modes in Gaussian formalism equate to direct sums of moments and to effect partial tracing one can simply delete moments of the traced mode.
If a state is sent through an attenuating channel that also injects thermal noise it undergoes the transformationρ = tr e Û ηρ ⊗ρ thÛη , whereρ th is a thermal state with average photon numbern th /(1 − η) and the mixing of two modes is caused byÛ η = exp iθ(â †â th +â † thâ ) which is a two-mode rotation operator with mixing angle θ = cos −1 √ η that also models the beamsplitter with η as a transmissivity. For the Gaussian state that has undergone lossρ G , we simply need to transform its moments µ where tr e is the partial trace over the environment noise mode (which here denotes simply removing appropriate rows and columns), S η as the symplectic beamsplitter transform and Σ th the covariance of a thermal mode with average photon numbern th /(1 − η), which has zero mean µ th = 0.
As a simple example illustrating the above, a one mode thermal Gaussian state withn has Wigner function W(x) = 2 π(1+2n) exp − |x| 2 2n+1 and covariance matrix Σ = (n + 1 2 )1 2 . When sent through an attenuating channel that contains thermal noise of mean photon number corresponding to a Wigner function of that is, a broadened Gaussian ifn th >n (1−η). In addition, the original state has its average photon number attenuated by a factor η.

III. DIRECT PHOTODETECTION
The direct photodetection measurement consists of just two outcomes: click or no-click, in other words the detector fires or it does not. Usually in quantum information, the measurement outcome is the expectation value of a POVM operator, so that, for a stateρ measured by an imperfect click detector with a dark count probability (the probability that the detector fires if no light falls upon it) provided by a thermal distribution withn d > 0 and quantum efficiency η < 1 [33], is the no-click probability. The operatorΠ × is the no-click operator andΠ = 1 −Π × is the click operator corresponding to a dark count click probability with no light incident on the detector ofn d /(1 +n d ). By this construction, all click probabilities can be defined in terms of their complementary no-click probabilities. Furthermore, giving the dark count probability distribution a thermal Gaussian character is the essential property that allows the analysis to be performed simply in the Gaussian framework.
Non-thermal dark count models are possible, such as Poisson statistics [34,35], but all valid models amount to measurement via projection on to a mixed state. Only for photon number resolving detectors would the detailed probability distribution of dark counts be noticeable.
Of course, the perfect click measurement occurs whenn d = 0 and η = 1, with vacuum The vacuum state is a Gaussian state with covariance matrix Σ = 1 2 1, so the optimal no-click detection of a Gaussian stateρ G depends simply on the overlap between the state and the vacuum, derived using Eqns. (4) and (5). For the more general result that includes nonzero dark noise and non-unit quantum efficiency -instead of converting Eq. (12) into a Wigner function we simply decompose measurement of the state by the imperfect click detector into preattenuation of the state measured by a perfect click detector (see Fig. 2), as these provide the same outcome where the prime indicatesρ G has passed through the loss channel in Eq. (7) that is easier to implement using the continuous-variable formalism presented in Eqns. (8).

IV. CONDITIONED SINGLE MODE STATES
The TMSV is an entangled Gaussian state which exhibits photon number and quadrature correlations (with zero-mean field). It has the following Fock basis wavefunction [36] containing modes i -idler and s -signal corresponding to different beams of light propagating, in principle, in different directions. The factor λ = n 1+n contains r as the squeezing amplitude in the form of the single-mode average photon numbern := sinh 2 r. Each beam or arm of the TMSV contains an averagen photons.
The two outcomes of click detection on the idler mode of the TMSV herald two different conditional signal states. When the no-click outcome occurs, the remaining signal state is where VET denotes the vacuum enhanced thermal state andρ i, s = |Ψ Ψ| i, s . The normalisation is simply the no-click probability from Eq. (14) of the idler mode: N = Pr(×|ρ I ), which may contain terms relating to the dark-noise and quantum efficiency depending on the optimality of the heralding measurement.
When a click occurs, the remaining signal state is conditioned to becomê which is the VST -vacuum suppressed thermal state. The VET state is a thermal state with mean photon number lower thann, the unconditioned mean. It has statistics similar to the partially traced TMSVρ s = tr iρi, s (see Fig. 3), which is a thermal state with covariance matrix Σ s = (n + 1 2 )1. The VST state is not thermal, but it can be expressed by Eq. (18) as a weighted difference of the unconditioned thermal state and the VET state.
The photon number distributions of the average and the conditioned states are shown in Fig. 3. It is clear from this figure that the no click result conditions the signal to be in a state with a smaller mean photon number, the VET. Consequently the click result conditions the signal to be in a state with a higher mean photon number, the VST. It is this increase in photon number that we can exploit for object detection. If optimal heralding detection is performed, with a perfect photodetector, the two conditioned signal states becomeρ vet → |0 0| andρ vst → 1 n k (n 1+n ) k |k k| − |0 0| -the latter state is completely vacuum removed. In such an ideal case, the vacuum removal increases the average photon number of the remaining signal mode by 1.
The Wigner functions of the conditional states can be found from the covariance matrix of the TMSV, that has sub-matrices V i, s = n + 1 2 1 2 and C = n(1 +n)1 2 for its quadrature correlations. This hollow covariance matrix is in a so-called standard form and can always be achieved from an arbitrary covariance matrix through a sequence of local or global rotation and squeezing symplectic transformations [37]. By performing a partial trace integral in phase space, we can extract the covariance matrix of the VET as which resembles a Schur complement matrix, indicating also that the vacuum enhanced thermal state is Gaussian. All of the states that we require can be represented as Gaussian weighted by the no-click probability from the idler mode N . This expression is useful for calculating the probabilities in the next section, as well as for demonstrating that non-Gaussian states can sometimes be written as weighted sums of Gaussian states.

V. QUANTUM ILLUMINATION FOR TARGET DETECTION
Consider a scenario where we send a single-mode stateρ as a probe to sense the presence of an object with reflectivity κ (0 < κ < 1) that is in a thermal noise bath. The probe has an average photon numbern and the noise is modelled by the thermal stateρ b with mean thermal photon numbern b . Conditioned by the presence or absence of the object, we Negativity of the Wigner function is exhibited only by non-classical states [38].
eventually have to discriminate between two different possible return stateŝ whereρ 0 is the thermal background stateρ b withn b that our detector receives if the object is absent;ρ 1 is the state we receive if the object is present as it contains reflected signal photons. Notice thatρ 1 is the same transformation as Eq. (7) with η replaced by κ and thatρ b has scaled average photon numbern b /(1 − κ).
A click detection measurement will satisfy the following: if the object is absent, the detector will fire with what could be called a false alarm probability dependent on the thermal background independent of any object properties κ or signal photon numbern. There is no object to detect so the signal does not reach the detector, hence this result applies to all possible sent signal states. The factors η andn d are the receiving detector efficiency and dark noise.
If the object is present, then sending a coherent state signalρ α that hasn = |α| 2 , µ = { √ 2n, 0} T and Σ = 1 2 1 2 gives a click probability forρ 1 If the target object is present and we are instead using the heralded TMSV as our signal state, then the single-mode state sent becomes one of the two conditioned states caused by click-heralding of the idler. If our local detector does not fire, then the receiving detector has a click probability of where the double prime on the covariance indicates the application of two loss channels: once through the object, once again for detector losses. Its explicit expression is which contain the idler detector efficiency η i and dark noisen di . If the local idler detector fires, then we send the stateρ vst , which enhances the click probability of the receiving detector with a probability of where V s = 1 2 is the thermal covariance of the vacuum suppressed state.
Plots of click probabilities for a detector receivingρ 0 vs.ρ 1 caused by sending different states can be found in Fig. 5. The left column of plots shows the probability of obtaining a count as a function of mean signal photon number. In most cases the background photon number is much higher than that of the signal. The no-object probabilityρ 0 is a flat line as it does not depend on the signal photon number. There is a noticeable heralding gap between the probability obtained using a coherent signal state and the vacuum suppressed state derived from a heralded TMSV of the same mean photon number. The advantage is largest for lown as heralding has a greater effect, suggesting that quantum illumination is advantageous for low average signal photon number. The effect persists even for higher loss scenarios, but in all cases the gap closes asn increases and eventually the coherent state outperforms the vacuum suppressed state. The reason for this is that the heralding has little effect on the thermal distribution for highn and the Poissonian nature of the coherent state photon number distribution peaks there. This effect dominates over the weak heralding effect.
The right column of plots shows the probability that we would assign to the presence H 1 or absence H 0 of an object based on a click at the receiving detector as a function of mean signal photon number. We assume no prior knowledge of the presence of the object.
A click at the detector increases the probability that the object is present and decreases the probability that it is absent. Again there is an gap at low mean photon number between heralded TMSV and coherent light, showing the quantum illumination advantage. Again the coherent state wins out as the mean photon number of the sent signal state increases.

VI. CLICK PROBABILITY MATCHING
The previous section shows clearly the single-shot advantage of quantum illumination for object detection with click detection, but it is not the only advantage that only allowing click detection confers. We can also exploit the fact that the average signal photocount distribution has particular forms, which specific detector click probabilities. The specific forms of distribution allow the coherent state to defeat quantum illumination for higher mean signal photon numbers in Fig. 5, but we can turn them to our advantage. Rather than comparing the results obtainable with classical coherent and heralded TMSV states of the same mean photon number, we can compare sent states that would give the same click probability at a detector. Without the extra heralding information it will not be easy to tell the difference between them. Such states are effectively indistinguishable by single detector clicks at Geiger-mode detectors, whatever their specific photon probability distributions.
The matching of click probabilities will not compromise source discoverability should we wish to illuminate covertly, rather it will enhance the covertness.
When one mode of the TMSV is observed independently of the other, it appears to be in a thermal state with average photon numbern s = sinh 2 r. If a click detector wishes to Pr nn - intercept the signal before interaction with the object it will fire with probability which is lower than the coherent state probability with average photon numbern α = |α| 2 = n s Instead of the above choice we can choosen s to match these click probabilities by equating the above two expressions which means thatn We are able to increase the mean photon number of the quantum illumination beyond that of the coherent state, without compromising source discoverability by a click detector (see Fig. 6). Whilst this is not strictly a quantum advantage it does more than offset the classical advantage that the coherent signal state has at higher signal photon numbers, as can be seen by comparing the red curves and the grey curves in both the top panel (click probabilities) and bottom panel (object presence or absence probabilities) of Fig. 6. The heralding advantage dominates at low mean photon numbers, where the potential for click probability matching advantage is limited (the difference between the orange dashed and red curves in the top panel, or the grey and blue in both panels). At signal photon numbers around one the click probability matching advantage begins to dominate.
One objection to click probability matching might be that we are limiting the means of discoverability to clicks made at a single detector. Multiple coincident detector clicks would allow the two different sources to be discriminated so, of course, the objection is correct.
However, this objection applies to distinguishing the coherent state from any state that does not have a Poissonian photon number probability distribution. Here we are specifically considering limited click detection as it is the simplest and most likely form in optics. Moreover, the probability of multiple coincident clicks in a real object identification system operating near the quantum level will be tiny. For such systems click probability matching can only decrease the chances of discoverability.

VII. MODELLING A SEQUENTIAL DETECTION PROCESS
As the small changes in the posterior probabilities from the no-information values of 1/2 in the presence of the target. In order to achieve greater confidence in estimation of the object's presence (or absence) we can apply the above process repeatedly in a multi-shot scenario.
We herald multiple sequential TMSV states and send them to interact with a possible target before repeated click detection. Experimentally this can be realised by sending a train of light pulses to probe the region of interest. As shown in Fig. 5, after each detector result given the object presence (absence), the posterior probability is updated from prior probabilities. We are able to simulate a repeated update of the estimate of the probability Pr(H 1 ) that an object is present, based on click or no-click measurement outcomes at both the idler and measurement detectors. The overall process is simple to simulate numerically and requires only Bayes' Law and a (pseudo)random number generator.
Suppose that we send a set of sequential states (shots) to the possible target object, making M sequential measurements. We begin by assuming that the prior probability of object presence is Pr(H 1 ) = 1/2, which can be considered as the zeroth trial Pr (m=0) (H 1 ).
We want to simulate the entire detection process either when a target object is present or when one is not. First assume that an object is present. In order to model the outcome of the single click-detector iteratively, we throw a uniformly-distributed random number p between 0 and 1 for each m-th measurement (0 ≤ m ≤ M ) and subject it to the following update rule . (33) Essentially, if the random number is smaller than the click probability we infer that the detector has fired. If the random number is larger than the click probability we infer that the detector has not fired. We update the probabilities of object presence or absence accordingly.
The updated estimated probability for object absence, which complements that above, is where Pr(ρ 1 | ) is the posterior probability for the state and detection outcomes calculated via Bayes' Law. The above example is with the object is present, as we have used Pr( |ρ 1 ) in the conditional statement. In order to simulate the sequential measurement given that the object is absent, we can switch the probability in the conditional statement to Pr( |ρ 0 ).
As production of the vacuum suppressed state requires a heralding click detector, we must adapt the procedure by subjecting a second random number, h to the following condition before moving on to calculate the posterior probabilities such that when no-click occurs on the idler we proceed to calculate conditional probabilities usingρ VET . If click heralding occurs we useρ VST . Eventually, after a number of measurements, the estimated probability will reach convergence, as shown by Fig. 7, after which we can conclude that the object is present or absent. We have chosen this simple convergence criterion here. We could, of course use other detection criteria, such as setting a probability threshold or looking at the differences between possible evolution of trajectories in the presence or absence of target objects.
The left panel of Fig. 7  The traces show a significant amount of noise but it seems clear that the heralded traces provide a much more stable, quicker detection and that it is better to click match. The smoother curves in grey, blue and red are the averages of 3000 trajectories. As an example of the utility of heralded TMSV states we use these to examine when the probability of object presence passes 0.8. The TMSV does this in less than half the number of shots of the coherent state and the click-matched TMSV in about a fifth of the number of shots.
The right panel in Fig. 7 shows the object present probability traces, this time when a target is not present to determine the monitoring detector statistics. The same click count distribution produces all traces, but the fact that which signal state was sent is known in each case allows a different updating of the probability. Similar advantages to the left panel are shown in excluding the presence of the object for the heralded TMSV and the click-matched heralded TMSV.

VIII. CONCLUSIONS
In this paper we have described a theory of quantum illumination for target detection in a noisy background. Our theory is written in terms of the the formalism of Gaussian quantum optics and can be wholly characterised using thermal Gaussian states, even in cases where the heralded state has a negative Wigner function and cannot be written as a Gaussian state. The theory is frequency-independent, so applies equally-well to lidar in the optical frequency range and to radar at microwave or radio wave frequencies, although the technical challenges of detectors sensitive to single photons are an issue here, as is the background rate at which they would fire. Research on such detectors is ongoing [39][40][41].
We have used our theory to show that the use of quantum illumination to provide a click-heralded state of TMSV of low average signal mean photon number provides a clear advantage, compared to using coherent states of the same mean photon number, for object detection under lossy, high background noise conditions. The return signal, under certain scenarios, is shown to be significantly more distinguishable from background noise. It provides enhanced click probabilities and, in turn, enhanced posterior probabilities useful for hypothesis testing in the multi-shot scenario. The detection of objects based on click-counts is also more stable and definitive.
The quantum illumination advantage is a direct consequence of the photon number correlations between two spatially separate modes of the TMSV, even though each individual beam has mean photon numbern. If the heralding detector has high quantum efficiency and a low mean dark count probability, characterised byn d , the average increase in the mean photon number of the signal beamn s is 1. This is a much more prominent effect for ρ vst produced from a low average photon number TMSV. When the idler clicks, the photon number distribution in the signal arm shifts away from the vacuum leading to higher click probability. This contrasts with the coherent state which becomes more vacuum-like at low mean photon number.
At higher mean photon number the heralding effect is much smaller and the coherent state provides better discrimination than the heralded TMSV. This is because the heralding has less effect on the TMSV and the coherent state probability distribution is more sharplypeaked around its mean value. We can, however, recover and increase the advantage by using a heralded TMSV with a higher unheralded mean photon number. This may appear to be cheating, but it can be accomplished by matching the detector click probabilities of the TMSV and coherent signal states, rendering them effectively indistinguishable to a Geiger-mode detector. This extra advantage might more correctly be termed a thermal state advantage, as it would also exist for illumination with classical single mode thermal states, but the combination of this and the heralding means that the quantum illumination can always outperform classical illumination.
Immediate advantages of quantum illumination using click-detectors are the readily available cost-effective equipment applicable to lidar systems, as well as he possibility of covert enhancement of photon number. Yet, the main drawback for performing quantum illumination experimentally would beρ vst production at the lown regime, as heralding probability for the TMSV dwindles quickly to zero vian 1+n . Ideally we would require a reliable entangled source, so that we could run the heralding process at high frequency to provide a sufficient rate of heralded quantum state production. The output of a laser is much easier to use for illumination. There are, however, other advantages to using quantum illumination: we have much more control over our state. The optical field has more degrees of freedom to be exploited, spatial, timing and polarisation to name but three. Also, more complicated measurements can be used to condition the signal state, which may turn out to be significantly more advantageous than simple click detection. The most basic example is that of multiple click heralding [42]. We will explore each of these in future work.