How Discord underlies the Noise Resilience of Quantum Illumination

The benefits of entanglement can outlast entanglement itself. In quantum illumination, entanglement is employed to better detect reflecting objects in environments so noisy that all entanglement is destroyed. Here, we show that quantum discord - a more resilient form of quantum correlations - explains the resilience of quantum illumination. We introduce a quantitative relation between the performance gain in quantum illumination and the amount of discord used to encode information about the presence or absence of a reflecting object. This highlights discord's role preserving the benefits of entanglement in entanglement breaking noise.

The absence of entanglement, however, does not necessarily imply classicality. Quantum protocols that operate with negligible entanglement exist [10,11], motivating the search for quantum resources beyond entanglement. Quantum discord stands as a prominent candidate [12][13][14]. Initially proposed to isolate the 'quantum' component of mutual information between two physical systems, discord is conjectured to be a potential quantum resource, responsible for the advantage of quantum processors [15]. While promising advances have been made in understanding the operational significance of discord [16][17][18][19][20][21][22], this remains a topic of significant debate. Contrary to entanglement, which is difficult to synthesize, discord is non-zero for almost every mixed state [23] and its practical merit contradicts the preconception that 'quantum' effects are fragile.
The consequences in connecting quantum discord and quantum illumination are two-fold. On the one hand, it reveals why quantum illumination efficiently works in a regime where entanglement does not survive; on the other hand, it establishes discord as a resource of immediate practical value. In this article, we establish this connection, demonstrating how discord directly quantifies the advantage of quantum illumination versus conventional classical strategies. In fact, despite entanglement being broken by the environment, discord sur- In conventional illumination, a single probe is sent into a noisy region to detect the presence of a potential object. (a) If the object is present, there is a small chance a reflected signal is detected; otherwise (b) the probe is completely lost and Alice just sees only random noise. In quantum illumination (c-d), Alice prepares two maximally-entangled systems, one is kept (idler) and the other sent for target detection (signal). The reflected signal and idler are finally detected by a joint measurement. Surprisingly, the use of an entangled source yields better performance, even though entanglement fails to survive the return trip. vives and is harnessed to preserve information about the potential presence of a reflecting object that would otherwise be lost. Remarkably, we find that the amount of discord associated with the sensing of the target coincides exactly with the performance gain of quantum illumination over the best conventional technique.
The General Scenario. Illumination aims to discern whether a low reflective object is present or absent in a distant region of intense noise (see Fig. 1). This can be viewed as a arXiv:1312.3332v1 [quant-ph] 11 Dec 2013 task in information retrieval. A distant region of space contains a bit of information that dictates the presence (x = 0) or the absence (x = 1) of the reflecting object. From this point of view, the goal of illumination is to retrieve the value x of a random variable X = {x, p x } with binary alphabet x ∈ {0, 1} and uniform distribution p 0 = p 1 = 1/2.
In the conventional approach, Alice probes the distant region with a suitable quantum system (where suitable implies a system that the reflector would potentially reflect), and monitors for a potential reflection. Let the probe be a d-dimensional quantum system, i.e, a qudit, in a pure state Φ = |φ φ|. If the reflector is absent (x = 1), the entirety of Φ is lost and Alice retrieves random environmental noise described by a maximally mixed state ρ E = d −1 I where I is the identity operator.
Otherwise (x = 0), the reflector may reflect the object back at Alice; and the noise ρ E Alice observes is biased by the signal Φ with some small weighting η 1. Thus, probing the the reflector corresponds to encode x ∈ {0, 1} into the output codewords By detecting the reflected qudit, Alice has a limited ability to distinguish these states and, therefore, to infer the value of x.
In quantum illumination, Alice improves her strategy by resorting to quantum correlations. She prepares a maximallyentangled state Ψ AB = |ψ AB ψ| of two qudits A and B, where |ψ AB = d −1 k |k A ⊗ |k B , with {|k } being an orthonormal basis. Then, she probes the target with the signal system A while retaining the idler system B in a quantum memory (or just a delay line in experimental settings). Now we have that x is encoded in the two codewords where ρ B = Tr A (Ψ AB ) represents the reduced (maximallymixed) state of the idler if the signal is completely lost.
In either approach, Alice ends up in possession of one of two potential codewords, ρ (0) or ρ (1) , depending on x. The better Alice can discriminate between these codewords, the better she can decode x. Quantum illumination thus outperforms its conventional counterpart when it is easier to distinguish ρ c , for any conventional input Φ.
The Advantage of Quantum Illumination. In order to measure the performance of illumination, we consider the Shannon distinguishability [24](SD) , also known as the accessible information. The motivation is that both the Shannon distinguishability and discord are quantified in units of bits, and thus facilitate our ultimate aim of direct comparison. The binary uniform variable X = {x, p x } describing the presence or absence of the target is encoded into an ensemble of two equiprobable codewords ρ (0) and ρ (1) . Their Shannon distinguishability SD(ρ (0) , ρ (1) ) represents the maximum amount of information which is retrievable on the target-variable X given an optimal quantum measurement on the ensemble (see Methods for more details). AB . In (b), the complete loss of signal is represented by a SWAP operation S between the signal system and environment. In either scenario, the resulting quantum channel is entanglement-breaking. The two scenarios can combine into a single circuit (c), composed of two sequential stages. In stage (i), environmental noise is injected into the signal arm by mixing in ρE, resulting in the state ρ (0) AB . In (ii), the presence or absence of the target can be modeled as encoding the binary variable X onto ρAB by applying S x , where S 0 = I is the identity.
The performance of quantum illumination is then given by AB ). On the other hand, the optimal performance achievable in conventional illumination is provided by maximizing the Shannon distinguishability of the output codewords ρ This advantage can be non-zero in scenarios where η 1, and ρ E is completely mixed. The reason behind this improved performance is not immediately clear, since the aforemention conditions imply that ρ (0) AB and ρ (1) AB are both highly entropic and completely separable, despite the original input Ψ AB being maximally entangled. Indeed, we can recast quantum illumination into a functionally equivalent quantum circuit, where the action of the noise is separated from that of the reflecting object (see Fig. 2). Irrespective of whether the reflector is present, the noise decoheres Alice's input Ψ AB into the separable state ρ (0) AB (cf. Fig. 2.c). Now the presence or absence of the target, i.e., the value x of the random variable X, is encoded into the state by applying the operator S x to the signal system, with S being the swap operator between the signal and environment (cf. Fig. 2.b). From this viewpoint, it is clear that the mechanism of quantum illumination is based on the remaining quantum correlations which survive in the separable state ρ (0) AB . These quantum correlations are quantified by discord and provide the resilient memory which stores the information about the presence or absence of the target.
The Role of Discord. Formally, the discord of the signalidler system, denoted as δ(A|B), quantifies the discrepancy between two types of correlations [14]. The first type is the quantum mutual information I(A : B) which accounts for the total correlations between the two systems A and B. As we have previously said, quantum illumination is able to work thanks to the discord δ(A|B) present in the separable state ρ (0) AB , which is the effective state responsible for sensing the target according to the equivalent quantum circuit in Fig. 2. However, in order to explicitly prove the role of discord as a physical resource for quantum illumination, we need to establish a direct connection between discord and the quantum advantage ∆I for the resolution of the target.
Consider the global state ρ represents the amount of discord which is associated with the knowledge of x. Equivalently, it represents the amount of discord which is expended for encoding x , i.e., storing the information about the target, in the effective state ρ AB . For instance, if there were no encoding of the target, i.e., target certainly present (p 0 = 1) or absent (p 1 = 1), then we would have δ enc (A|B) = 0. Furthermore, we note that δ enc (A|B) can be positive only if ρ (0) AB has non-zero discord δ(A|B) > 0. The advantage of quantum illumination is its unique access to the quantity δ enc (A|B). As we show in the Methods, the advantage of quantum illumination coincides exactly with the discord expended for encoding x, that is ∆I = δ enc (A|B). ( As a result, quantum illumination gives an advantage ∆I > 0 only if the effective state ρ has non-zero discord, and the corresponding quantum advantage ∆I is directly provided by the amount of discord δ enc (A|B) which is expended for resolving the target.
Thus, quantum discord is identified as the underlying physical resource for quantum illumination, providing an extra memory in which information on the target is robustly stored. While entanglement does not survive in quantum illumination, the survival of discord is essential for it to have any advantage over conventional illumination.
A Simple Example. We illustrate the equivalence of Eq. (2) in the case where signal and idler are two-level quantum systems, i.e., qubits. The environment is flooded with random qubits, such that ρ E = I/2. For example, this may model the detection of a multi-faceted, rotating, object in noise [1].
The conventional approach probes the target with a pure state |φ , returning either ρ and I q , respectively, are plotted versus the target reflectivity η in Fig. 3.a. The difference between these curves (shaded region) quantifies the gain ∆I of quantum illumination.
Then, as we can see from Fig. 3.b, the state of the system after noise, ρ (0) AB , is always separable for sufficiently small values of η. Nevertheless, ρ (0) AB contains discord, part of which can be harnessed to store information about x. In comparing Fig. 3.a with Fig. 3.b, we see that the amount of discord expended for resolving the target δ enc (A|B) coincides exactly with the advantage ∆I of quantum illumination.

DISCUSSION
In this article, we have shown that discord is the underlying resource responsible for the performance advantage of quantum illumination. In maximally noisy environments, discord can survive when entanglement cannot. Quantum illumination exploits these surviving quantum correlations to encode extra information regarding the potential presence of a reflective object. The amount of discord used to encode this information is shown to coincide exactly with the enhanced performance of quantum illumination, the equivalence holding for systems of arbitrary dimensions. This connection addresses a two-fold purpose. It helps explain why benefits of entanglement may survive entanglement-breaking noise; and complements the recent interpretations of discord as a physical resource establishing its crucial role in quantum technology.
For simplicity, in our paper we have considered a uniform probability distribution p 0 = p 1 = 1/2 for the random variable X describing the status of the target. In the supplementary information, we relax this assumption and we consider a binary variable X with an arbitrary distribution. The performance advantage of quantum illumination ∆I still satisfies the equality ∆I = δ enc in this completely general case. Thanks to this versatility, our result can be applied to many variants of the orginal quantum illumination protocol. For instance, a prominent candidate is the adoption of quantum illumination as a cryptographic communication protocol [8,9]. In such a variant, one assumes that Bob lives in the region of intense noise, and uses the reflector to distribute a secretkey with Alice. More generally, this could be achieved by the application of appropriate unitaries, as in the original cryptographic protocols based on two-way quantum communication [25]. The application of our techniques would then connect the improved security of these protocols with the discord which is employed to encode the secret information.
In general, quantum illumination is one among a broad collection of protocols aimed at determining certain properties of unknown quantum channels. There is quantum channel dis-crimination [26], where a discrete set of channels encodes a multivariate random variable, to be retrieved by optimizing input probes and output detections. A simple example is the binary case of quantum loss detection, where one has to determine if a quantum channel is lossy or not. Then, there is quantum metrology, where the set of channels is continuous and one typically seeks to measure the unknown parameter θ of a unitary e iθH . In each of these protocols, numerical links between discord and performance has led the proposals that the former may be a resource for the latter [15,27,28]. Our techniques could formalize these ideas through direct analytical relations, leading to a unified understanding of how the benefits of entanglement survive when entanglement dies.

METHODS
The performance of illumination can be quantified directly in units of bits by adopting the Shannon distinguishability [24]. A binary random variable X = {x, p x } can be encoded into an ensemble of codeword states {ρ (x) , p x } of a quantum system A. In order to retrieve information back from the ensemble, one has to perform a general positive operator value measurement (POVM) M on A, whose output defines another random variable K M . The information retrieved is the mutual information between X and the measured output K M , i.e., I(X : The maximum accessible information is the Shannon distinguishability of the codewords SD(ρ (0) , ρ (1) ), which is defined by maximizing I(X : K M ) over all POVMs, i.e., SD(ρ (0) , ρ (1) ) = max M I(X : K M ). ( While this quantity generally has no closed form, it coincides with the Holevo information [29,30] when the two codewords commute [ρ 0 , ρ 1 ] = 0, i.e., whereρ = p x ρ (x) is the average state of system A, and S(·) is the von Neumann entropy. In general, for noncommuting states [ρ 0 , ρ 1 ] = 0, the Holevo information is known to provide an upper bound.
In our paper, we assume the worst-case scenario where the presence or absence of the target are equiprobable events, so that the variable X has uniform distribution p 0 = p 1 = 1/2. (This condition is assumed for both simplicitly and similarity with the original quantum illumination protocol. However, it is not necessary in general, since our derivation continues to hold for binary variables with arbitrary distributions.) We then quantify the advantage of quantum illumination in terms of improved Shannon distinguishability of the quantum scheme over the best possible conventional approach, i.e., we consider AB ) and I max c ), where the maximization is taken over all pure states Φ of a single qudit.
In choosing an information theoretic quantifier (Shannon distinguishability), we are adopting a measure of performance which is different from that introduced by Lloyd [1], where quantum illumination was quantified by the probability of guessing x correctly. Both measures carry operational meaning. In gambling scenarios where Alice bets double or nothing on x, for example, the probability of success measures the chance that Alice wins each round; whereas the amount of information Alice gains about x dictates Alice's expected earnings using Kelly's optimal betting strategy [31]. The performance measures are closely related: Knowledge of one bounds the other from both above and below, and the scaling properties of the two measures coincide [24]. In using an information theoretic quantifier, we have followed an approach similar to that of quantum reading [32], where the mutual information was used to better characterize the optimal readout of a classical memory.
Outline of the Proof. The key idea behind our proof is to first impose an additional restriction on the quantum illumination protocol. In standard quantum illumination, Alice could use arbitrary measurements to distinguish between ρ (0) AB and ρ (1) AB . Here, we impose a specific measurement procedure. Alice is required to first make a local measurement on the idler B, followed by a local measurement on the signal A. This naturally restricts the amount of information she can access about X to some maximal value I c . We then establish that: (i) The optimal performance of quantum illumination, subject to this restriction, coincides with the best performance using conventional illumination I c = I max c .
(ii) The loss in performance in using this third approach over quantum illumination is I q − I c = δ enc (A|B).
The detailed proofs of (i) and (ii) are below. Together, they imply that Eq. (2) must hold. Proof of Statement (i). Recall that in the conventional approach, Alice probes for the reflecting object with a pure qudit