One-Shot Detection Limits of Quantum Illumination with Discrete Signals

A minimally-invasive way to detect the presence of a stealth target is to probe it with a single photon and analyze the reflected signals. The efficiency of such a conventional detection scheme can potentially be enhanced by the method of quantum illumination, where entanglement is exploited to break the classical limits. The question is, what is the optimal quantum state that allows us to achieve the detection limit with a minimal error? Here we address this question for discrete signals, by deriving a complete and general set of analytic solutions for the whole parameter space, which can be classified into three distinct regions, in the form of phase diagrams for both conventional and quantum illumination. Interestingly, whenever the reflectivity of the target is less than some critical values, all received signals become useless, which is true even if entangled resources are employed. However, there does exist a region where quantum illumination can provide advantages over conventional illumination; there, the optimal signal state is an entangled state with an entanglement spectrum inversely proportional to the spectrum of the environmental state. These results not only impose fundamental limits in applications such as quantum radars, but also suggest how to become immune against the attack of minimally-invasive detection.

Introduction-One of the most important tasks in quantum information science is to understand how physical procedures related to information processing can be improved by exploiting quantum resources such as entanglement [1]. Apart from the well-established applications such as quantum computation [2], simulation [3,4], teleportation [5], metrology [6], etc., the area of quantum illumination [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] is emerging as a promising and novel quantum method for increasing the sensitivity or resolution of target detection in a way that can go beyond the classical limits. The primary goal of quantum illumination is to detect the presence or absence of a target, with potentially a low reflectivity and in a highlynoisy background, by sending out an entangled signal and performing joint (POVM) measurements. More specifically, the setup of quantum illumination consists of three parts: (i) a source emits a signal entangled with an idler system kept by an receiver; (ii) if a target exists, the receiver obtains the reflected part of the signal in addition to the background noise; otherwise, only the background noise can be received; (iii) the receiver perform a joint POVM measurement on the whole quantum system and infer from it the presence of the target.
An intriguing feature of quantum illumination is that it is highly robust against loss and decoherence; one can still gain quantum advantages, even if the signal is applied to entanglement-breaking channels [22]. As an important application, one can apply quantum illumination to secure quantum communication [9,[23][24][25][26], where the sender encode a 0-or-1 message by controlling the presence of absence of an object and the receiver determine its presence by illuminating entangled photons; in this way, an eavesdropper who does not have access to another half of the entangled signal could virtually know nothing about the message communicated [9]. An experimental implementation [24] of the protocol above suggested that quantum illumination can provide a reduction up to five orders of magnitude in the bit-rate error against an eavesdropper attack. Furthermore, experimental implementations of quantum illumination have been extended from the optical domain [11,24,27] to the microwave domain [12]. This progress is significant, as in the optical domain, the natural (thermal) background radiation on average contains less than one photon per mode. Consequently, artificial noise is necessary to implement quantum illumination at optical wavelengths [24].
In fact, quantum illumination represents an applications of a larger class of problems called quantum channel discrimination [28]. However, only very few analytic solutions have been discovered; in fact, quantum channel discrimination is generally a very hard computational problem [29]; it is complete for the quantum complexity class QIP (problems solvable by a quantum interactive proof system), which has been shown [30] to be equivalent to the complexity class PSPACE (problems solvable by classical computer with polynomial memory).
Here we show that the problem of one-shot quantum illumination, for any given parameter regime and for signals with any finite dimension, can be solved completely with a compact analytic solution. More specifically, our main results include a derivation of an analytic expression for the minimized error probability for target detection in quantum illumination, where the minimization is over all possible POVM measurements and for all possible finite-dimensional (entangled) probe states. Furthermore, the optimal state we obtained depends only on the spectral information of the environment signal; in other words, the minimized error probability can al-ways be achieved without even knowing the reflectivity and occurrence probability of the target.
On the other hand, quantum discord, a measure of nonclassical correlation [31], was suggested [15,32] to be the reason for the quantum advantages gained by quantum illumination. However, this conclusion is not applicable to our results. In fact, the authors [32] only consider completelymixed environment; one can construct counter examples violating the conclusion of Ref. [32] for general environments (see appendix).
Model of one-shot quantum illumination-Let us first consider conventional illumination. Suppose the individual photonic state ρ be described by an d-dimensional density matrix, and the thermal noise of the environment is denoted by, if the target is absent, the probe signal ρ is completely lost; we can only receive the noisy state from the environment, i.e., (ii) even if the target is present, the detection may not be perfect; the reflecting portion of the signal is quantified by the reflectivity, η ∈ [0, 1], and the quantum channel is, For quantum illumination, the probe signal is entangled with another subsystem, and the quantum channels are applied partially, i.e., (i) when the target is absent, where ρ B ≡ tr A ρ AB , and when it is present, The problem of target detection with quantum illumination can be regarded as a problem of quantum channel discrimination [28]: given a pair of quantum states ρ 0 and ρ 1 , associated with probabilities p 0 and p 1 , p 0 + p 1 = 1. In connection with the problem of quantum illumination, we should take ρ 0 = E 0 (ρ) and ρ 1 = E 1 (ρ) for conventional illumination, and ρ 0 = (E 0 ⊗ I)(ρ AB ) and ρ 1 = (E 1 ⊗ I)(ρ AB ) for quantum illumination. We are interested in finding the input states that can minimize the minimum error, given by [33], where A ≡ tr √ A † A denotes the trace norm of a matrix A. Here, the corresponding trace norms are labeled by Ω c (ρ) and Ω q (ρ AB ) , respectively for conventional and quantum illumination, where and In other words, the corresponding minimum errors are given by P c,q err ≡ 1 2 (1 − Ω c,q (ρ) ). Our ultimate task is to optimize over all possible states, i.e., for quantum illumination, where P c err,♦ ≡ max ρ AB P c err and P q err,♦ ≡ max ρ AB P q err , or explicitly, When the two subsystems are uncorrelated, i.e., ρ AB ≡ ρ A ⊗ ρ B , the quantum case is reduced to the conventional case; therefore, it is necessarily true that quantum illumination is not worse than conventional illumination, i.e., P q err P c err and P q err, P c err, . Finally, we note that both the values of p 0 (and p 1 ), and the reflectivity η can be determined in the beginning by state tomography.
Main results-Our major results contain a family of complete analytic solutions for both conventional and quantum illumination for any d-dimensional signal state and any given environmental state ρ E . For both conventional and quantum illuminations, the minimal-error probabilities are strongly dependent on the the occurrence probabilities {p 0 , p 1 } and the reflectivity η of the target. In general, we can divide the parameter space into three distinct regions, namely (I,II,III).
(Region I): (i) p 0 < p 1 , and (ii) η < η * ≡ 1 − p 0 /p 1 . For both conventional and quantum illuminations, the minimal error is given by, Furthermore, the optimal strategy for both quantum and conventional illumination does not even require a measurement of the signals; one can simply guess "yes" (present of the target) for all cases. As whenever p 0 < p 1 , the error for this simple strategy is equal to p 0 , i.e., P err = p 0 . We summary this result as follows (the proof is left in the appendix): Result 1 (Region I, for both conventional and quantum illuminations). (a) the minimal errors for conventional and quantum illumination are equal to p 0 , i.e., P err = p 0 , and (b) the bound can be achieved with any (pure or mixed) state.
The minimal error is given by, for both conventional and quantum illuminations. Moreover, both η c → 0 and η q → 0 vanishes as λ d → 0, which implies that region II vanishes for both cnventional and quantum illuminations. The same performance is achieved by guessing "no" (i.e., absence of the target) for all events. Here which is related to the harmonic mean of of the eigenvalues {λ i } of environmental signal ρ E . Note that for λ d > 0, it is always true that λ h is always less than the smallest eigenvalue λ d of ρ E , i.e., . Therefore, the region II for the case of quantum illumination is always smaller than that of conventional illumination. (see Fig. 1). To summarize (see proof in appendix), we have Result 2 (Region II for conventional and quantum illumination). (a) the minimal error for conventional and quantum illumination is equal to p 1 , i.e., P err = p 1 , and (b) the bound can be achieved with any (pure or mixed) state.
(Region III): (the region excluded by region I and II) For conventional illumination, the minimal error over all possible input states is given by, and for quantum illumination, Here the parameter, γ ≡ p 1 (1 − η) − p 0 , depends on the occurrence probabilities {p 0 , p 1 } of the target and the reflectivity η. In this region, γ < 0 is negative and a decreasing function of η, which implies that both P c err, and P c err, decrease with the increase of the reflectivity η. Furthermore, the difference between the classical and quantum cases (i.e., quantum advantage) depends on the difference, λ d − λ h , i.e., For conventional illumination, the input state that can minimize the detection error is given by the eigenstate |θ d of ρ E associated with the smallest eigenvalue λ d . To summarize (see proof in appendix), we have Result 3 (Region III: minimal error decreases with reflectivity η for conventional illumination). The minimal error P err over all possible conventional input states is given by P c which is obtained by choosing |ψ = |θ d to be the eigenvector of ρ E associated with the smallest eigenvalue.
To understand the result (Eq. (12)) of the minimal error for quantum illumination in Region III, we summarize the steps for achieving it below: Sketch of the proofs for quantum illumination-The main physical quantity to be investigated is: and Furthermore, we express the bipartite pure state in the follow- where the vectors |u i 's are not assumed to be normalized. In general, they are nonorthogonal to one another. However, since the eigenvalues |θ i 's are orthonormal, the normalization condition implies The next task is to bound the minimum eigenvalue, E g ≡ λ min (H q ) of the matrix H q . The corresponding eigenvector |g ψ , where H q |g ψ = E g |g ψ , can always be expanded by the the following vectors (in a way similar to |ψ ), |g ψ = d i=1 |θ i |v i , where, again, the vectors |v i 's are neither normalized nor orthogonal to one another. We found that (see appendix) the eigenvalue is minimized when we choose |u i = |v i for all i's, which means that |g ψ = |ψ . Finally, we found that the minimum eigenvalue, E g = λ h − α, of H q can be achieved by choosing an input of the form (see appendix), where µ i = λ h /λ i ; this result is summarized as follows: Result 4 (Optimal state for quantum illumination). The lower bound, λ h − α, of E g can be achieved by the input Below, we provide three different examples to illustrate our results.
Example 2: binary signals Let us consider the case where the signals are two-dimensional, which means that ρ E is a 2 × 2 Hermitian matrix. In its diagonal basis (labeled as {|0 , |1 }), we write ρ E = λ0 0 0 λ1 . Since the trace norm is invariant under unitary transformation, we can always choose to have the pure state to be optimized as follows: |ψ = µ 0 |0 + µ 1 |1 . where both parameters, µ 0 ≥ 0 and µ 1 ≥ 0, are positive, and The eigenvalues λ ± of Ω c are given by The trace norm of Ω c is given by one of the following possibility: Note that det Ω c is a product of the two eigenvalues; the condition of det{Ω c } > 0 implies that either both eigenvalues are positive or both negative. Example 3: completely-mixed environment Suppose the returning signal from the noisy environment is completely mixed, i.e., ρ E = I/d, the corresponding matrix Ω c (|ψ ψ|), for any pure state |ψ , can be diagonalized explicitly to give Ω c (|ψ ψ|) = p 1 η + γ d + d−1 d |γ|. In region I, where p 0 ≤ 1 2 and η ≤ η * ≡ 1 − p 0 /p 1 , we have γ ≡ p 1 (1 − η) − p 0 ≥ 0. As a result, Ω c = |p 1 η + γ| = |p 1 − p 0 | and hence P c err = p 0 . On the other hand, in region II, p 0 > 1 2 (where γ < 0) and η < ( p0 p1 − 1) 1 d−1 (where p 1 η + γ/d < 0), we have again Ω c = |p 1 η + γ| = |p 1 − p 0 |, but it gives P c err = p 1 . In region III, Ω c = p 1 η + (2/d − 1) γ, which gives P c err = p 1 (1 − η) − γ/d. Complementary results-If for all the events, we simply guess 'yes' (i.e., presence of the target) whenever p 0 < p 1 and 'no' (i.e., absent) whenever p 0 > p 1 . Then, the error for guessing wrong is given by the minimum of the probabilities p 0 or p 1 , i.e., min {p 0 , p 1 }. For example, the instance shown below shows that the number of wrong decisions (i.e., 'yes' when the object is absent '0') is equal to the number of absent events '0'. This error bound is relevant to the cases where the reflectivity η is zero, i.e., ρ 0 = ρ 1 for both conventional and quantum illumination, which gives P err = 1 2 (1 − |p 0 − p 1 |). In other words, when there is no signal related to the absence/presence of the target, the best strategy one can make to minimize the error of discrimination is exactly the strategy mentioned above.
Another interesting question is how the reflectivity η affect the error bound. Intuitively, we would believe that the higher the value of η, the smaller the error bound. This intuition can be justified by the following theorem (proof in the appendix): Result 6 (Monotonicity of minimal error). For a given reflectivity η, and density matrix ρ, and the minimal error given by, P err (η) = 1 The minimal error is a non-increasing function of the reflectivity, i.e., if η η , then P err (η) P err (η ).
On the other hand, the optimization can be taken over pure states only (see proof in appendix).
Conclusions-In this work, we presented complete solutions to the problem of one-shot minimum-error discrimination for both conventional and quantum illuminations, for finite-dimensional signals. The analysis is divided into three regions. Region I are the same for both conventional and quantum illumination; the minimal error is a constant and does not depend on the reflectivity of the targetthe optimal strategy is achieved via simple guess. The same is similar for region II, except that using quantum illumination can shrink the boundary of region II. For region III, quantum illumination can yield a lower minimal error than conventional illumination. Result: Region I for both classical and quantum illuminations Suppose then (a) the minimal errors for conventional illumination and quantum illumination are equal to p 0 , i.e., and (b) the bound can be achieved with any (pure or mixed) state.
Proof. If p 0 ≤ 1/2 and η ≤ η * , we have γ 0, which also implies that the Hermitian matrix, Ω c(q) , is a positive sum of two density matrices (with positive eigenvalues).
Consequently, all eigenvalues λ i = i| Ω c(q) |i , with an eigenvector |i , are positive, i.e., λ i 0. In this case the trace norm of Ω c(q) can be obtained directly by taking the trace, i.e., which implies the result stated in Eq. (17). Note that the whole argument is applicable to any pure state.
Result: Region II for conventional illumination Suppose (i) p 0 1/2, and (ii) η −η * ( λmin 1−λmin ) , with λ min = λ min (ρ E ) 0, the minimal eigenvalue of the environment signal ρ E , then (a) the minimal error for conventional illumination is equal to p 1 , i.e., and (b) the bound can be achieved with any (pure or mixed) state.

RESULT FOR REGION III
For conventional illumination in region III, we have γ < 0. Therefore, we can always write for some α > 0. We shall see that (i) the trace norm of Ω c is determined by the minimum eigenvalue of the matrix and (ii) the smallest eigenvalue is minimized by choosing the signal state as the eigenstate |e k with the minimal eigenvalue. These results come from the following lemmas.
Proof. Consider the eigenvalue equation for the same matrix, i.e., (ρ − α |ψ ψ|) |e k = E k |e k , for any k ∈ {1, 2, 3, .., d}, which can be written as or |e k = α(ρ − E k I) −1 |ψ ψ| e k . Now, as ψ| e k = 0, we therefore have the following relation: Therefore, in terms of the eigenvalues λ k and eigenvectors θ k of ρ = k λ k |θ k θ k |, the eigenvalues E k are the roots of the equation.
Now, the right-hand side increases monotonically from zero as E increases from −∞ to zero. Therefore, depending on the value of α, there can be, at most, one negative eigenvalue for the matrix ρ − α |ψ ψ|.
The same result can be derived in an alternatively way, as follows.
Lemma 2 (Positivity of eigenvalues II). Let ρ be a ddimentional density matrix, let |ψ be a pure quantum state, and α > 0 a real number. Then there is at most one negative eigenvalue of the operator ρ − α |ψ ψ|.
Proof. Suppose we can find two distinct negative eigenvalues such that E d < E d−1 < 0. Consider the subspace spanned by the corresponding eigenvectors, V = Span {|e d , |e d−1 }, of the matrix ρ − α |ψ ψ|. Clearly, there exists a linear combination, denoted by which is orthogonal to |ψ , i.e., ψ ψ ⊥ = 0. Then, ρ − α|ψ ψ| maps |ψ ⊥ to ρ|ψ ⊥ , i.e., On the other hand, when restricted to the subspace V , the operator ρ − α|ψ ψ| is negative definite, since it has its all eigenvalues negative. Explicitly, This conclusion contradicts the fact that ρ is a density matrix, which must be positive semidefinite. There exists at most one negative eigenvalue. Finally, if |ψ happens to be an eigenvector, then ρ|ψ = (E + α) |ψ . Therefore E −α, which means E can be negative, zero, or positive.
Lemma 3 (Problem of eigenvalue minimization). Following the previous lemma, if E d ≤ 0, then the minimum error P err is minimized by minimizing the negative eigenvalue E d of the matrix ρ E − α |ψ ψ| with fixed ρ E and α.
Proof. Let us now express the minimal error as where α = p 1 η/|γ|. Denote E k 's as the eigenvalues of the matrix ρ E − α |ψ ψ|. Then, we have where we have applied the result of the previous lemma. Now, we have k =d E k = 1 − α − E d and hence which depends linearly with the smallest eigenvalue E d ; the more negative E d is, the smaller P err becomes.
As a result, the minimum eigenvalue E d of the matrix ρ E − α |ψ ψ| is bounded by λ min − α E d . Furthermore, this bound can be saturated by choosing |ψ = |θ d .
Therefore, for region III, we will only need to consider the minimum error resulted from sending the eigenstates of ρ E with the minimum eigenvalue.
Theorem (Region III: minimal error decreases with reflectivity η). The minimal error P err over all possible conventional input states is given by which is obtained by choosing |ψ = |θ d to be the eigenvector of ρ E associated with the smallest eigenvalue.
Proof. Let us consider the matrix, Here we consider the range where but p 1 η + γλ d > 0. Note that tr(Ω c ) = p 1 η + γ is a sum of the d − 1 negative eigenvalues and one positive eigenvalue, p 1 η + γλ d . Therefore, the trace norm is obtained by Ω c = −tr(Ω c ) + 2 (p 1 η + γλ d ), or Finally, as P c err = (1 − ||Ω c ||) /2, we have P c To check the consistency of the result above, when λ d = 0, we have Note that when γ = 0, then P c err = p 0 . Moreover, when we have p 1 η = −γλ d , which implies that P c In the special case where p 0 = p 1 = 1/2 and d = 2, γ = −η/2. Therefore, in agreement with the example.

RESULTS FOR QUANTUM ILLUMINATION
Lemma 5 (Lower bound of eigenvalue value). Given a density matrix ρ E and a pure state |ψ , the minimum eigenvalue (associated with the eigenvector |g ψ ), E g ≡ λ min (ρ E ⊗ ρ B − α |ψ ψ|), is bounded below by Proof. Let us consider the explicit form of the smallest eigenvalue : we can obtain an upper bound of ψ |g ψ by (i) taking absolute values for each term, i.e., By defining x i ≡ | u i |v i |, we obtain a lower bound for the smallest eigenvalue Next, we are going to minimize the lower bound of E g , subject to a constraint, where C 1 is not greater than unity as | ψ |g ψ | 1. We found that E g is minimized when α > λ h where we have proportional to the harmonic mean of the set of eigenvalues {λ i } and λ h λ min . On the other hand, when α λ H , the smallest eigenvalue of H Q is positive, i.e., λ min 0. In this case, the trace norm is equal to the trace, i.e., H q = trH q .
Region II of quantum illumination Lemma 6 (Minimization with Lagrange multiplier). The Proof. Let us introduce a Lagrange multiplier l m . The the lower bound of E g , labeled by is minimized when the condition, holds, where g ≡ d i=1 x i − C. Explicitly, we have ∂f ∂x i = 2(λ i x i − αC), ∂g ∂x i = 1 , which gives x i = (αC + l m /2) /λ i , and hence C = d i=1 x i = (αC + l m /2) ( d i=1 λ −1 i ). As a result, the minimal value of f is given by, Therefore, When α λ h ≡ ( d i=1 λ −1 i ) −1 , f is minimized by choosing C = 0, which implies all of the eigenvalues of H q are positive, i.e., λ i (H q ) 0. However, when α > λ h , the lower bound f is minimized by setting C 2 = 1, which implies E g λ h − α.
Note that the condition of α λ h is equivalent to This defines the region II of quantum illumination. There, all the eigenvalues of Ω q has the same sign. Similar to the classical case (see theorem 2), the minimal error is given by which can be achieved by any state. Recall that the region II for conventional illumination is bounded by and is equivalent to the trivial strategy. Quantum illumination is capable of shrinking the boundary to as λ h ≤ λ min .

Region III of quantum illumination
Result: Optimal state for quantum illumination The lower bound, λ h − α, of E g can be achieved by the input state, where µ i = λ h /λ i .
Proof. Let us consider the following ansatz, |ψ = d i=1 µ i |θ i |θ i , where the amplitudes µ i 's are assumed to be non-negative (µ i 0) and normalized ( d i=1 µ 2 i = 1). The expectation value, H q = ψ| H q |ψ = ψ| ρ E ⊗ρ B |ψ −α, is given by We can achieve the lower bound, H q = λ h − α, by setting µ 2 i = λ h /λ i . (One can readily check that the normalization condition is obeyed as

COMPLIMENTARY RESULTS
Result: Upper bound of minimal error The error probability P err is bounded above by either p 0 or p 1 , i.e., Proof. Mathematically, this result comes from the definition of the trace norm. Let us consider the case of p 0 ≥ p 1 first. Denote e k 's as the eigenvalues of the operator p 0 ρ 0 − p 1 ρ 1 . Furthermore, we label those non-negative eigenvalues as e + k ≥ 0 and the negative ones as e − k < 0. In this way, we can write the trace norm as the difference between these two set of eigenvalues, i.e.,