Discriminating Strength: a bona fide measure of non-classical correlations

A new measure of non-classical correlations is introduced and characterized. It tests the ability of using a state {\rho} of a composite system AB as a probe for a quantum illumination task [e.g. see S. Lloyd, Science 321, 1463 (2008)], in which one is asked to remotely discriminate among the two following scenarios: i) either nothing happens to the probe, or ii) the subsystem A is transformed via a local unitary R_A whose properties are partially unspecified when producing {\rho}. This new measure can be seen as the discrete version of the recently introduced Intereferometric Power measure [G. Girolami et al. e-print arXiv:1309.1472 (2013)] and, at least for the case in which A is a qubit, it is shown to coincide (up to an irrelevant scaling factor) with the Local Quantum Uncertainty measure of D. Girolami, T. Tufarelli, and G. Adesso, Phys. Rev. Lett. 110, 240402 (2013). Analytical expressions are derived which allow us to formally prove that, within the set of separable configurations, the maximum value of our non-classicality measure is achieved over the set of quantum-classical states (i.e. states {\rho} which admit a statistical unravelling where each element of the associated ensemble is distinguishable via local measures on B).


I. INTRODUCTION
In recent years strong evidences have been collected in support of the fact that composite quantum systems can exhibit correlations which, while not being accountable for by a purely classical statistical theory, still go beyond the notion of quantum entanglement [1]. In the seminal papers by Henderson and Vedral [2], and Ollivier and Zurek [3], this new form of non-classicality was gauged in terms of a difference of two entropic quantities -specifically the quantum mutual information [4] (which accounts for all correlations in a bipartite system), and the Shannon mutual information [5] extractable by performing a generic local measurement on one of the subsystems. The resulting functional, known as quantum discord [2], enlightens the impossibility of recovering the information contained in a composite quantum system by performing local detections only. It turns out that this intriguing feature of quantum mechanics is not directly related to entanglement [6]. Indeed, even though all entangled states are bound to exhibit non-zero value of quantum discord, examples of separable (i.e. non entangled) configurations can be easily found which share the same property -zero value of discord identifies only a tiny (zero-measure) subset of all separable configurations [7]. In spite of the enormous effort spent in characterizing this emerging new aspect of quantum mechanics, a question which is still open is whether and to what extent the new form of quantum correlations identified by quantum discord can be considerd as a resource and exploited to give some kind of advantage over purely classical means. Due to the variety of contexts where quantum theory proved to be a useful tool for developing new technological ideas (such as information theory, thermodynamics, computation and communication), this gave rise to a number of alternative definitions and quantifiers of discord-like correlations, see e.g. [1] and references therein. This proliferation stems also from the difficulty of identifying a measure which is at the same time operationally well defined and easily computable, even for the case of a two qubit system. As a paradigmatic example, let us recall the geometric discord [8] which can be effortlessly computed at the price of being increasing under local operations [9].
In an effort to simplify the theoretical scenario, in this paper we introduce a new measure of quantum correlations, the Discriminating Strength (DS), which turns out to be a valid tradeoff between computability and the fulfillment of the criteria that every good discord quantifier should satisfy [10]. Most importantly, it also possesses a clear operative meaning, being directly connected with the quantum illumination procedures introduced in Refs. [11][12][13][14]. Being the counterpart of the recently introduced Interferometric Power for continuos variable estimation theory [15], the DS enlightens the benefit gained by quantum state discrimination protocols when general quantum correlations, not necessarily in the form of entanglement, are employed. Finally, we provide a formal connection between our new measure and the Local Quantum Uncertainty Measure (LQU) introduced in Ref. [16] whose operational meaning was not yet completely understood. Specifically we show that LQU is a special case of DS when the state is used as probe to determine the application of a local unitary which is close to the identity. Furthermore, for qubit-qudit systems one can verify that LQU and DS always coincide up to a proportionality factor.
The manuscript is organized as follows. In Sec. II we introduce a paradigmatic state discrimination scheme and we quantify how good a generic state ρ can perform in the discrimination. In Sec. III we show that the same quantifier satisfies all the properties required for a bona fide measure of discord. Moreover we present the connection between our measure and the LQU measure and we provide some simple analytical formulas for some special cases (specifically pure states and qubit-qudits systems). In Sec. IV we focus on the set of separable states and we determine the maximum value of the DS on this set in the qubit-qudits case. In order to formally introduce our new measure of nonclassicality it is useful to recall the Quantum Chernov Bound (QCB) [17]. This is an inequality which characterizes the asymptotic scaling of the minimum error probability P (n) err,min (ρ 0 , ρ 1 ) attainable when discriminating among n-copies of two density matrices ρ 0 and ρ 1 [17]. By optimizing with respect to all possibile Positive Operator Valued Meausure measurements (POVM) aimed to distinguish among the two possible configurations, and assuming a 50% prior probability of getting ρ ⊗n 0 or ρ ⊗n 1 , one can write [18] P (n) err,min := the optimal detection strategy being the one which discriminates among the negative and non-negative eigenspaces of the operator ρ ⊗n 0 − ρ ⊗n 1 . For large enough n, the dependance of the error probability on the number of copies can be approximated by an exponential decay P (n) err,min (ρ 0 , ρ 1 ) e −n ξ(ρ0,ρ1) =: Q(ρ 0 , ρ 1 ) n , (2) characterized by the decay constant Accordingly, the larger is Q(ρ 0 , ρ 1 ) the less distinguishable are the states ρ 0 and ρ 1 . The limit in (3) corresponds to the QCB bound [17] and reads which implies Furthermore if at least one of the two quantum states ρ 0 or ρ 1 is pure, then QCB reduces to the Uhlmann's fidelity [19], i.e.
Let us now consider the following quantum illumination scenario [11][12][13][14]. A first party (Alice) prepares n copies of a density matrix ρ of a bipartite system AB composed by a probing component A and a reference component B, while a second party (the non-cooperative target Robert) selects an undisclosed unitary transformation R A from a set S of allowed transformations. Next Alice sends her n subsystems A to Robert who is allowed to do one of the following actions: induce the same rotation R A on each of the n subsystems A, or leave them unmodified -see Fig. 1. Only after this step Robert reveals . . .

Robert Alice
FIG. 1: (Color online): sketch of the discrimination problem discussed in the text. (1) A first party (say Alice) prepares n copies of a bipartite state ρ of a composite system AB and (2) sends the probing subsystems A to a second party (say Robert) while keeping the reference subsystems B on her laboratory. (3) Robert can now decide whether or not a certain unitary rotation RA he has previously selected from a set S of allowed transformations, should be applied (locally) on each one of the probes A. (4) After this action the subsystems A are returned to Alice and the chosen RA is revealed to her. By exploiting this information and by performing the most general measure on her systems, she has now to determine which option (i.e. the application of RA or the non application of RA) Robert has selected. the chosen rotation R A and sends back the A subsystems. Alice is now requested to guess whether the rotation R A has been implemented or not, i.e. to discriminate between ρ ⊗n 0 = ρ ⊗n (no rotation) and ρ ⊗n 1 = (R A ρR † A ) ⊗n (rotation applied). For this purpose of course she is allowed to perform the most general POVM on the n copies of the transformed states. In particular, as in a conventional interferometric experiment, she might find useful to exploit the correlations present among the probes A and their corresponding reference counterparts B [it is important to stress however that, due to the lack of prior info on R A , Alice cannot perform any optimization with respect to the choice of her initial state ρ]. In this scenario we define the "discriminating strength" of the state ρ by quantifying Alice's worst possible performance through the quantity where the maximization is performed over the set S of allowed R A , and where the symbol A → B enlightens the different role played by the two subsystems in the problem -an asymmetry which is a common trait of the majority of non-classical correlations measures introduced so far [1]. From Eqs. (4) and (7) it is clear that the higher is D A→B (ρ) the better Alice will be able to determine whether a generic element of S has been applied or not to A. It is a natural guess to expect that the capability shown by the input state ρ of recording the action of an arbitrary local rotation, should increase with the amount of correlations shared between the probe A (which has been affected by the rotation) and the reference B (which has not). This behavior would be analogous to the one displayed by the Interferometric Power measure discussed in [15], which quantifies the worst-case precision in deter-mining the value of a continuous parameter. Clearly the choice of S plays a fundamental role in our construction: for instance allowing S to coincide with the group U A of all possible unitary transformations on A, including the identity, would give D A→B (ρ) = 0 for all states ρ. To avoid these pathological results we find it convenient to identify S with the special family of R A parametrized as where H Λ A is an Hamiltonian of assigned non-degenerate spectrum represented by the elements of the diagonal matrix with λ 1 > λ 2 > ... > λ d A (d A being the dimension of the system A) and λ 1 − λ d A < 2π (a condition the latter which can always be enforced by properly relabeling the entries of Λ). Accordingly we have where now U A spans the whole set U(d A ). For each given choice of Λ (8) we thus define the quantity the maximization being performed over the set {H Λ A } of the Hamiltonians of the form (9). In the next section we will show that, for all given choices of the spectrum Λ the functional (11) fulfills all the requirements necessary for attesting it as a proper measure of non-classical correlations [1].

III. PROPERTIES
In this section we show that the discriminating strength (11) is a bona fide measure of non-classicality. We also clarify the connection between our measure and the LQU measure introduced by Girolami et al. in Ref. [16]. Finally we provide close analytical expressions that, in some special cases, allow one to avoid going through the cumbersome optimization over the set {H Λ A } of the Hamiltonians (9).
with p i being probabilities, {|i A } being an orthonormal basis of A and {ρ B (i)} being a collection of density matrices of B (these are the only configurations for which it is possible to recover partial information on the system by measuring A, without introducing any perturbation [1]); 2. it is invariant under the action of arbitrary local unitary maps, W A and V B on A and B respectively, i.e.
3. it is non-increasing under any completely positive, trace-preserving (CPT) [20] map Φ B on B; 4. it is an entanglement monotone when ρ is pure. Proof: Being R Λ A endowed with a non-degenerate spectrum, this is equivalent to stating that ρ and H Λ A are diagonal in the same basis {|i A } of H A , and thus ρ reduces to a CQ state of the form (12).
2) First note that for every unitary operator U it holds has the same spectrum of H Λ A so that the maximization domain in (11) remains unchanged along with the maximum value.
3) This follows from the very definition of the QCB. Indeed, the minimum error probability in (1) is achieved by optimizing over all possible POVMs (Positive-Operator Valued Measurements) on (AB) ⊗n . Any local map Φ B on B commutes with the phase transformation determined by H Λ A , and thus can be reabsorbed in the measurement process. This modified measurement is at most as good as the optimal one, implying that the asymptotic error probability, and hence Q, cannot decrease. This We will prove that if a pure state |ψ is transformed into another pure state |φ by LOCC (Local Operations and Classical Communication), then D Λ A→B (|φ ) ≤ D Λ A→B (|ψ ). We remind that, due to the purity of the input and output states, a generic LOCC transformation which maps the vector |ψ in |φ can always be realized via a single POVM on A followed by a unitary rotation on B conditioned by the measurement outcome, see e.g. [20]. In other words, we can write where , and {V j B } is a set of unitary operators on B. Introducing the set of probabilities (14) it follows that for all j corresponding to p j = 0 we must have Observe also that for each H Λ A , there exists an H Λ B which has the same components in the Schmidt basis of |ψ , that is From Eq. (6) it follows then that for pure input states the maximization over all H Λ A is equivalent to a maximization over all H Λ B . This allows to write where weH Λ B labels the Hamiltonian for which the maximum is reached. Along the same lines, we have where the second identity follows from Eq. (15) by absorbing the unitary operator V jB into the maximization over H Λ B . The rhs of the latter expression can be bounded from above by noticing that the maximum of a given function is greater than the function evaluated at a given point. In particular we have whereH Λ B has been introduced in Eq. (18). Finally, applying the Cauchy-Schwarz inequality we get hence concluding the proof.

B. A formal connection between DS and LQU measures
The LQU measure of non-classical correlations was introduced in Ref. [16]. Given a state ρ of the bipartite system AB it can be computed as where (23) is the Wigner-Yanase skew information [21] and where, as in Eq. (11), the maximum is taken over the set {H Λ A } of the Hamiltonians (9). A connection between (22) and our DS measure follows by taking a formal expansion of Eq. (11) with respect to Λ, i.e.
where in the third identity we used the following property.
Lemma 1: Given ρ a density matrix and Θ = Θ † a Hermitian operator we have where {c } are the eigenvalues of ρ which have being organized in decreasing order (i.e. c ≥ c for ≤ ). Equation (24) establishes a formal connection between our DS measure and the LQU measure, providing hence a clear operational interpretation for the latter. Specifically the LQU can be seen as the DS measure of a discrimination process where Λ is a small quantity, i.e. where the allowed rotations R Λ A of Eq. (10) are small perturbation of the identity operator. As we shall see in Sec. III E, the relation among DS and LQU becomes even more stringent when A is a qubit system: indeed, in this special case, independently from the dimensionality of B, the two measure are proportional.
C. Dependence upon Λ According to Sec. III A all choices of the matrices Λ as in Eq. (8) provide a proper measure of non-classicality for the states ρ. Even though one is tempted to conjecture that the case where Λ has an harmonic spectrum (i.e. λ k − λ k−1 = const for all k = 2, 3, · · · , d A ) should be somehow optimal (i.e. yield a more accurate measure of non-correlations), the relations among these different DSs at present are not clear and indeed it might be possible that no absolute ordering can be established among them (this is very much similar to what happens for the LQU of Ref. [16]). Here we simply notice that since QCB is invariant under constant shifts in the local Hamiltonian spectrum, i.e. Q(ρ, , for all incoming states ρ and for b ∈ R, we can always add a constant to Λ at convenience without affecting the corresponding DS measure, i.e.

D. Discriminating strength for pure states
Let |ψ be a pure state of AB with Schmidt decomposition [20] given by where ρ A = Tr B [|ψ A ψ|] is the reduced state of |ψ on H A . From the spectral decomposition (9) of H Λ A , one can perform the trace in (28) over the eigenbasis of Λ and get where now the maximization is performed over the set of the double stochastic matrices M with elements M (k|j) = A λ k |U † A |j A j|U A |λ k A . We remind that according to the Birkhoffs theorem [22] M can be written as a convex combination of permutation matrices Π α (corresponding to the permutation π α ), i.e.
Therefore, we can rewrite Eq. (29) as Note that if d B < d A , the number of Schmidt coefficients is smaller than the number of eigenvalues λ k . In this case, the expressions above hold as long as one considers the state (27) as having d A − d B Schmidt coefficients equal to zero, i.e. one must apply the permutations to the set By convexity it derives that the optimization over the set {p α } in (31) can be explicitly carried out by choosing probability sets {p α } which have only a single element greater than zero (and thus equal to 1), from which we finally derive where the maximization over the infinite set of Hamiltonians H Λ A required by its definition (see Eq. (11)) has been replaced by a maximization over the group of permutations {π α } on the set of the Schmidt coefficients q j .

Hamiltonians with harmonic spectrum
If the spectrum of the Hamiltonian H Λ A is harmonic with fundamental frequency ω = |λ i − λ i+1 | ≤ 2π/d A , Eq. (32) can further simplified. More precisely, let us relabel the set of eigenvalues {λ i } as where [x] stands for the integer part of the real parameter x. Let us also reorder the Schmidt coefficients of |ψ as q 1 ≥ q 2 ≥ . . . ≥ q d A (where again some of them must be set to zero if d B < d A ). By representing the phases e iλ k as unitary vectors in the complex space, one derives that the permutation π maximizing the sum in (32) is the one which associates q 1 to λ 0 = 0, q 2 to λ 1 = ω, q 3 to λ −1 = −ω, q 4 to λ 2 = 2ω, q 5 to λ −2 = −2ω, etc., yielding

E. Discriminating strength for qubit-qudit systems
We conclude the Section by considering the case in which subsystem A is given by a single qubit, and determine a closed expression for the discriminating strength.
Exploiting the gauge invariance (26) we set, without loss of generality, Λ = Diag{−λ, λ} and parameterize the set of local Hamiltonians acting on A as H Λ A = λn · σ A , wheren is a unit vector in the Bloch sphere and σ A = (σ A,1 , σ A,2 , σ A,3 ) is the vector formed by the Pauli operators. In what follows we will set σ (n) A =n · σ A . Under these hypothesis, the QCB can be written as where in the last passage we have used the fact that σ (n) A is Hermitian and Lemma 1 to conclude that the minimization in s is solved for s = 1/2 (see also Ref. [23], footnote 5 on page 11). Replacing this into Eq. (11) we finally obtain where is the LQU measure for a qubit-qudit system [16] see Eqs. (22) and (23). The identity (35) strengthen the formal connection between DS and LQU detailed in Sec. III B and provides a simple way to compute the DS for qubit-qudit systems. Indeed using the results of Ref. [16] it follows that with ξ max (W ) being the maximum eigenvalue of a 3 × 3 matrix whose elements are given by If ρ is pure, ρ = |ψ ψ|, the discriminating strength reduces to where q 1 and q 2 are the Schmidt coefficients of |ψ . In particular, notice that for separable pure states we have |q 1 − q 0 | = 1 and the discord vanishes (see property 1 in Sec. III). On the other hand, for maximally entangled qubit-qudit states we have q 0 = q 1 = 1/2 and the DS reaches the maximum value sin 2 λ (see property 4).

IV. MAXIMIZATION OF THE DISCRIMINATING STRENGTH OVER THE SET OF SEPARABLE STATES
The main role played by the discord in the realm of quantum mechanics is enlightening the presence of those quantum correlations which cannot be classified as quantum entanglement. Here, we investigate the behavior of the discriminating strength when computed on the set of separable states ρ (sep) (yielding zero entanglement). We will prove that for all qubit-qudit systems (d A = 2 and d B ≥ 2), the maximum discord over the set of separable states is reached over the subset of pure Quantum-Classical (pQC) states given by convex combinations of pure (non necessarily orthogonal) states {|ψ k A } on A and orthonormal basis {|k B } on B, i.e.
the {p k } being probabilities. For the case d B ≥ 3 we have an analytical proof of this fact, which allows us to solve the maximization and show that the following identity holds (see Sec. IV C).
A. pure-QC states maximize the DS over the set of separable states: case dB = ∞ A generic separable state can always be written as where {|ψ k A } are (possibly non-orthogonal) pure states on H A and {ρ By direct calculation, one can easily verify that the above inequality is saturated a pure-QC state ρ (pQC) of Eq. (40) obtained by replacing the density matrices ρ (43) with orthogonal projectors |k B k| (notice that this is possible because B is infinite dimensional). Indeed in this case we have Next we show that the maximum DS attainable over the set of pQC states (and hence over the set of separable states) cannot be larger than 2 3 sin 2 λ. To do so let us first consider the uniform pQC state ρ where we set H Λ (see Sec. III E) and introduced cos θ j =n ·r j .
In the limit d → ∞ the series d j=1 cos 2 θ j converges to an integral over the solid angle, which does not depend on the orientation ofn, i.e. Therefore we have To prove that the above quantity is also the maximum value of DS over the whole set of pure-QC states (40) we notice that, proceeding as in Eq. (48), we can write wheren * indicates the direction which is saturating the maximization. This vector is clearly a function of the state ρ (pQC) , i.e. it depends on the probabilities p j and on the vectorsr j . If we define the state ρ (pQC) R , obtained from ρ (pQC) by applying to the vectorsr j a rotation matrix R ∈ SO(3) , we have where the vector saturating the maximization in Eq. (51) now corresponds to Rn * . By introducing an ancillary system C, associated to the Hilbert space H C , and a set of N 3D-rotations {R k }, mapping each vertex of the regular N-polyhedron on all vertices (including itself), one can define the density matrix On the other handρ (pQC) ABC can be also arranged as where the density matrices ρ It is important to observe that since B is infinite dimensional, there always exists a stateρ (pQC) of AB which is fully isomorphic toρ (pQC) ABC , from which it follows where Q(ρ,n) := Q ρ, e iλσ (n) Thanks to expansion (53), we get from which, taking the maximum overn, it results Finally, since for all k, ρ (pQC) R k and ρ (pQC) share the same DS (see Eq. (52)), we get On the other hand, thanks to expansion (55) we have and therefore The above inequality is saturated in the limit N → ∞, where each ρ (pQC) u,N,j approaches the state ρ (see Eq. 50). We therefore have The identity (41)  If H B is finite dimensional we are not guaranteed about the possibility of mapping a generic separable state in the a pure-QC state. Thus relation (46) could be in principle violated. However by embedding H B into a larger system having infinite dimension one can still invoke the result of the previous subsection to say that To prove Eq. (41) it is hence sufficient to produce an example of a pure-QC state (40) that achieves such upper bound. Of course the sequence of uniform states (47) cannot be used for this purpose because now d B is explicitly assumed to be finite. Instead we take with |0 B , |1 B , |2 B being orthonormal elements of H B , which is a properly defined p-QC state whenever the dimension d B is larger than 3. As in the first line of Eq. (51), its associated discriminating strength can be then computed as, wherer j is the vector in the Bloch sphere of the state |ψ j while H Λ A = λσ (n) A . We are interested in the case where {r j } is an orthonormal triplet (i.e. the three vectors identifying three Cartesian axes in the 3D-space). Notice that this does not mean that the corresponding states are orthogonal: instead they are mutually unbalanced states (e.g. , so that (67) corresponds to an (unbalanced) Generalized B92 (GB92) state [24]. From the normalization condition on vectorn, it derives that the squared scalar products (n ·r j ) 2 define a set of probabilities, since j=0,1,2 (n ·r j ) 2 = |n| 2 = 1 .
Thus, the maximization involved in (68) can be trivially performed by choosingn parallel to ther j associated to the maximum weight p j . This gives By observing that for a three event process the maximum probability can never be smaller than 1/3, we conclude that the maximum DS over the set of GB92 states is achieved by the Equally Weighted (EW) one With this choice we get which shows that, also for d B finite and larger than 3, the upper bound (66) is achievable with a pure-QC state, hence proving (41).
C. p-QC states maximize the DS over the set of separable states: case dB = 2 (qubit-qubit) The argument used in the previous section cannot be directly applied to analyze the qubit-qubit case (i.e. d A = d B = 2), because for those systems the states (67) and (71) cannot be defined. Furthermore we will shall see that the upper bound (66) is no longer tight. To deal with this case we first consider the class of QC state and show the maximum of DS, equal to (1/2) sin 2 λ, is achieved on the set of pure-QC states. Then we resort to numerical optimization procedures to show that no other separable qubit-qubit state can do better than this, hence verifying the identity (42).

Maximum DS over QC states
A generic QC state for the qubit-qubit case can be expressed as with 0 ≤ φ ≤ π and 0 ≤ s i ≤ 1, which yields where φ 0 = 0, φ 1 = φ, R(θ) = exp −i θ 2 σ 2 and (76) We now have all the ingredients necessary for the computation of the matrix elements W αβ . Thanks to the orthogonality of |0 B and |1 B , this gives It derives that the eigenvalues of W reduce to Being W 22 < 1 and W 11 + W 33 = 1 + W 22 , we have that ξ + is the maximum eigenvalue. Therefore Eq. (37) yields where It derives the equality being saturated when W 22 = 0, W 13 = 0 and W 11 −W 33 = 0. The first condition sets to 1 the purity of τ 0 and τ 1 (s 2 0 = s 2 1 = 1), the second and third conditions imply φ = (2n + 1)π/2, with n ∈ Z, and p = 1/2. We conclude that the maximum of the DS on the set of QC states is achieved on B92-like states, which are pure-QC, that is being and sin(φ) = ±1 and |± = (|0 ± |1 )/2.

Separable qubit-qubit states: numerical results
We conclude our analysis by providing numerical evidence that (1/2) sin 2 λ is the maximum value reached by the discriminating strength over the all set of separable states as anticipated in Eq. (42). We recall that a generic separable state of two qubit systems can always be written as a finite convex sum of direct products of pure states for A and B [25], i.e.
with 1 ≤ N ≤ 4. We remark that here no orthogonality constraint has to be imposed on either sets of pure states {|ψ j A } and {|χ j B }, on H A and H B , respectively. The Bloch sphere formalism allows us to define, for all j |ψ j A ψ j | := I +û j · σ A 2 and |χ j B χ j | := I +v j · σ A 2 .
Summarizing, all qubit-quibit separable states are characterized by a set of N probabilities and 2N vectors of unit norms. The case N = 1 is trivial (all separable states are completely uncorrelated) and the DS is always zero. Therefore, we have numerically analyzed the cases N = 2, N = 3 and N = 4 and plot our results in Fig. 2 The details of this numerical analysis are presented in Appendix B.

V. CONCLUSIONS
In this paper we have introduced, under the name of discriminating strength, a novel measure of discord-like correlations, i.e. correlations that, even though not being addressable as quantum entanglement, are still nonclassical. In the mare-magnum of definitions and measures [1], each stemming from a different way in which quantum correlations can be used to outperform purely classical systems, the discriminating strength finds its natural collocation in the context of state discrimination. More precisely, it quantifies the ability of a given bipartite probing state to discriminate between the application or not of a unitary map to one of its two subsystems, when a large number of copies of the probing state is at disposal. We report that in a similar context, the noisy quantum illumination [12], a recent paper [26] has put forward a connection between the advantage yielded by quantum illumination over the best conceivable classical approach, and the amount of quantum discord (as in Ollivier and Zurek [3]) surviving in a maximally entangled state after the interaction with a noisy environment. Here however, our goal was to define a quantity which has a clear operative meaning (characterizing quantitatively each bipartite state as a resource for a specific task) and is also easy to compute, at least in some simple cases.
Specifically, we have proved that the discriminating strength fits all the requirements ascribing it as a proper measure of quantum correlations [1]. We have also provided a closed expression of this measure for some special cases, such as pure states and qubit-qudit systems. For the latter case we have also shown an explicit connection with another measure of quantum correlations, the local quantum uncertainty [16], which, in the most general case, can be seen to approximate the discriminating strength in the limit where the unitary map is close to the identity. Next, we have focused on the class of separable states and proved, by means of both analytical and numerical methods, that for all qubit-qudit systems the discriminating strength reaches its maximum on the set of pure quantum classical states. Finally, we have explicitly determined this maximum value.
To conclude, we remind that by definition the discriminating strength depends on the spectral properties of the encoding Hamiltonian H Λ A . In other words, for each specific choice of Λ one can in principle define a different measure of quantum correlations (a similar problem also affects the local quantum uncertainty). It would be therefore interesting to investigate if there exists a criterion for comparing different measures arising from different spectral properties of H Λ A . In this perspective, we believe that our analysis marks a further step towards a novel classification of a vast set of non-classicality measures. dinateŝ n 1 := (cos φ 0 , cos φ 1 , cos φ 2 ) , n 2 := (− cos φ 0 , cos φ 1 , cos φ 2 ) , n 3 := (cos φ 0 , − cos φ 1 , cos φ 2 ) , n 4 := (cos φ 0 , cos φ 1 , − cos φ 2 ) , have the same value of j=0,1,2p j cos 2 φ j but are associated to the following values for ∆(φ 1 , φ 2 , φ 3 ), with a = A cos φ 2 cos φ 1 , b = B cos φ 2 cos φ 0 and c = C cos φ 0 cos φ 1 . From F1 it derives that at least one of the vectorsn 1,2,3,4 will have ∆ positive.
We therefore conclude that max n j=0,1,2 p j (n ·r j ) 2 ≥ max n j=0,1,2p where the last identity follows from the fact that {cos 2 φ j } is a probability set, since it fulfills the normalization condition j=0,1,2 cos 2 φ j = 1, see Eq. (A3). Replacing this into Eq. (68) finally yields where the last inequality holds because the largest of three positive quantities summing to 1 cannot be smaller than 1/3.

Appendix B: Numerical analysis for qubit-qubit separable states
This appendix is devoted to discussing in deeper details the numerical analysis presented in Sec. IV C 2.
We have computed the discriminating strength of a two-qubit system in an arbitrary separable state, which, without loss of generality can be written as with 1 ≤ N ≤ 4, andû j ,v j normalized vectors in the Bloch sphere [25]. Let us start with the case N = 2. The set of probabilities {p i } can be labelled as {p 1 , p 2 } = C 2 {sin α, cos α} C 2 = 1 sin α + cos α , with 0 < α ≤ π/4. The latter constraint implies 0 < p 1 ≤ p 2 . Similarly, we have parametrized the unit vectorsû j andv j by means of the polar and azimuthal angles, 0 ≤ θ u,v j ≤ π and 0 ≤ φ u,v j < 2π, respectively. For each angle, we have taken a set of uniformly distributed values within the corresponding range, and perform all possible combinations. Finally, we have set some additional constraints in the numerical code in order get rid of those states which are equivalent under local unitary transformations. Thanks to this procedure, we have generated a set of ∼ 7 × 10 8 separable states and found that the state with maximum DS corresponds to the B92 state (82) with D Λ A→B = 1/2 sin 2 (λϕ), thus confirming what shown in Sec. IV C.
We have repeated the same analysis for the case N = 3 by setting {p1, p2, p3} = C 3 {sin α sin β, sin α cos β, cos α} C 3 = 1 sin α(sin β + cos β) + cos α (B3) with 0 < α, β ≤ π/4 to ensure that 0 < p 1 ≤ p 2 ≤ p 3 . We thus generated a set of ∼ 2 × 10 6 separable states. The maximum DS detected within this ensemble is ∼ 0.485 sin 2 (λϕ), and corresponds to α = 3π/16, β = π/4, θ u,v j = φ u,v j = 0, for j = 1, 2 Up to local unitary transformations, this set of parameters describes the state which is almost equivalent the B92 state (82) found for N = 2. We foresee that, by means of a finer graining of the parameter space, one should be able to include in the ensemble generated with this procedure the B92 state and reach 1/2 sin 2 (λϕ) as the highest value for DS.