Geometric quantum discord with Bures distance

We define a new measure of quantum correlations in bipartite quantum systems given by the Bures distance of the system state to the set of classical states with respect to one subsystem, that is, to the states with zero quantum discord. Our measure is a geometrical version of the quantum discord. As the latter it quantifies the degree of non-classicality in the system. For pure states it is identical to the geometric measure of entanglement. We show that for mixed states it coincides with the optimal success probability of an ambiguous quantum state discrimination task. Moreover, the closest zero-discord states to a given state are obtained in terms of the corresponding optimal measurements.


I. INTRODUCTION
One of the basic questions in quantum information theory is to understand how quantum correlations in composite quantum systems can be used to perform tasks that cannot be performed classically, or that lead classically to much lower efficiencies [1]. These correlations have been long thought to come solely from the entanglement among the different subsystems. This is the case for quantum computation and communication protocols using pure states. For instance, in order to offer an exponential speedup over classical computers, a pure-state quantum computation must necessarily produce multi-partite entanglement which is not restricted to blocks of qubits of fixed size as the problem size increases [2]. For composite systems in mixed states, however, there is now increasing evidence that other types of quantum correlations, such as those captured by the quantum discord of Ollivier and Zurek [3] and Henderson and Vedral [4], could provide the main resource to exploit in order to outperform classical algorithms [5][6][7][8] or in some quantum communication protocols [8][9][10][11]. The quantum discord quantifies the amount of mutual information not accessible by local measurements on one subsystem. One can generate mixed states with non-zero discord but no entanglement by preparing locally statistical mixtures of nonorthogonal states, which cannot be perfectly distinguished by measurements. The strongest hint so far suggesting that the discord may in certain cases quantify the resource responsible for quantum speedups is provided by the deterministic quantum computation with one qubit (DQC1) of Knill and Laflamme [12]. The DQC1 model leads to an exponential speedup with respect to known classical algorithms. It consists of a control qubit, which remains unentangled with n unpolarized target qubits at all stages of the computation. For other bipartitions of the n + 1 qubits, e.g. putting together in one subsystem the control qubit and half of the target qubits, one finds in general some entanglement, but its amount is bounded in n [13]. Hence, for large system sizes, the total amount of bipartite entanglement is a negligible fraction of the maximal entanglement possible. On the other hand, the DQC1 algorithm typically produces a non-zero quantum discord between the control qubit and the target qubits [5], save in some special cases [14]. This has been demonstrated experimentally in optical [6] and liquid-state Nuclear Magnetic Resonance [7] implementations of DQC1. This presence of non-zero discord can be nicely interpreted by using the monogamy relation [15] between the discord of a bipartite system AB and the entanglement of B with its environment E if ABE is in a pure state [16]. The precise role played by the quantum discord in the DQC1 algorithm is still, however, subject to debate (see [8] and references therein).
A mathematically appealing way to quantify quantum correlations in multi-partite systems is given by the minimal distance of the system state to a separable state [17]. The Bures metric [18,19] provides a nice distance d B on the convex cone of density matrices, which has better properties than the Hilbert-Schmidt distance d 2 from a quantum information perspective. In particular, d B is monotonous and Riemannian [20] and its metric coincides with the quantum Fisher information [21] playing an important role in high precision interferometry. As a consequence, the minimal Bures distance to separable states satisfies all criteria of an entanglement measure [17], which is not the case for the distance d 2 . This entanglement measure has been widely studied in the literature [22][23][24][25]. By analogy with entanglement, a geometric measure of quantum discord has been defined by Dakić, Vedral, and Brukner [14] as the minimal distance of the system state to the set of zero-discord states.
However, the aforementioned authors use the Hilbert-Schmidt distance d 2 , which leads to serious drawbacks [26].
The aim of this work is to study a similar GQD as in [14] but based on the Bures distance d B , which seems to be a more natural choice. This distance measure of quantum correlations has a clearer geometrical interpretation than other measures [17,27] based on the relative entropy, which is not a distance on the set of density matrices. We show that it shares many of the properties of the quantum discord. Most importantly, as in the description of quantum correlations using the relative entropy [27,28], our geometrical approach provides further information not contained in the quantum discord itself. In fact, one can look for the closest state(s) with zero discord to a given state ρ, and hence learn something about the "position" of ρ with respect to the set of zero-discord states. The main result of this paper shows that finding the Bures-GQD and the closest zero-discord state(s) to ρ is closely linked to a minimal error quantum state discrimination (QSD) problem.
The task of discriminating states pertaining to a known set {ρ 1 , . . . , ρ n } of density matrices ρ i with prior probabilities η i plays an important role in quantum communication and quantum cryptography. For instance, the set {ρ 1 , . . . , ρ n } can encode a message to be sent to a receiver. The sender chooses at random some states among the ρ i 's and gives them one by one to the receiver, who is required to identify them and henceforth to decode the message. With this goal, the receiver performs a measurement on each state given to him by the sender. If the ρ i are non-orthogonal, they cannot be perfectly distinguished from each other by measurements, so that the amount of sent information is smaller than in the case of orthogonal states. The best the receiver can do is to find the measurement that minimizes in some way his probability of equivocation. Two distinct strategies have been widely studied in the literature (see the review article [29]). In the first one, the receiver seeks for a generalized measurement with (n + 1) outcomes, allowing him to identify perfectly each state ρ i but such that one of the outcomes leads to an inconclusive result (unambiguous QSD). The probability of occurrence of the inconclusive outcome must be minimized. In the second strategy, the receiver looks for a measurement with n outcomes yielding the maximal success probability P S = n i=1 η i P i|i , where P i|i is the probability of the measurement outcome i given that the state is ρ i . This strategy is called minimal error (or ambiguous) QSD. The maximal success probability P opt S and the optimal measurement(s) are known explicitly for n = 2 [30], but no general solution has been found so far for more than two states (see, however, [31]) except when the ρ i are related to each other by some symmetry and have equal probabilities η i (see [29,32,33] and references therein). However, several upper bounds on P opt S are known [34] and the discrimination task can be solved efficiently numerically [35,36]. Let us also stress that unambiguous and ambiguous QSD have been implemented experimentally for pure states [37] and, more recently, for mixed states [38], by using polarized light.
Let ρ be any state of a bipartite system with a finite-dimensional Hilbert space. We will prove in what follows that the Bures-GQD of ρ is equal to the maximal success probability P opt S in the ambiguous QSD of a family of states {ρ i } and prior probabilities {η i } depending on ρ. Moreover, the closest zero-discord states to ρ are given in terms of the corresponding optimal von Neumann measurement(s). The number of states ρ i to discriminate is equal to the dimension of the Hilbert space of the measured subsystem. When this subsystem is a qubit, the discrimination task involves only two states and can be solved exactly [29,30]: P opt S and the optimal von Neumann projectors are given in terms of the eigenvalues and eigenvectors of the hermitian matrix Λ = η 0 ρ 0 − η 1 ρ 1 . In a companion paper [39], we use this approach to derive an explicit formula for the Bures-GDQ of a family of two-qubits states (states with maximally mixed marginals) and determine the corresponding closest zero-discord states.
This article is organized as follows. The definitions of the quantum discords and of the Bures distance are given in Sect. II, together with their main properties. In Sect. III, we show that the Bures-GQD of a pure state coincides with the geometric measure of entanglement and is simply related to the highest Schmidt coefficient. We explain this fact by noting that the closest zero-discord states to a pure state are convex combinations of orthogonal pure product states. The link between the minimal Bures distance to the set of zero-discord states and ambiguous QSD is explained and proved in the Sect. IV. The last section contains some conclusive remarks and perspectives. The appendix contains a technical proof of an intuitively obvious fact in QSD. Let us first recall the definition of the quantum discord [3,4]. The total correlations of the bipartite system in the state ρ are described by the mutual information I A:B (ρ) = S(ρ A )+S(ρ B )−S(ρ), where S(·) stands for the von Neumann entropy. The amount J B|A (ρ) of classical correlations is given by the maximal reduction of entropy of the subsystem B after a von Neumann measurement on A. Such a measurement is described by an orthogonal family where the maximum is over all von Neumann measurements {π A i }, q i = tr(π A i ⊗ 1 ρ) is the probability of the measurement outcome i, and ρ B|i = q −1 i tr A (π A i ⊗ 1 ρ) is the corresponding post-measurement conditional state of B. The quantum discord is by definition the difference δ A (ρ) = I A:B (ρ) − J B|A (ρ) between the total and classical correlations. It measures the amount of mutual information which is not accessible by local measurements on the subsystem A. Note that it is asymmetric under the exchange A ↔ B. It can be shown [40] that δ A (ρ) ≥ 0 for any ρ. Moreover, where is an orthonormal basis of H A , σ B|i are some (arbitrary) states of B depending on the index i, and q i ≥ 0 are some probabilities, i q i = 1. The fact that δ A (σ A-cl ) = 0 and from the non-negativity of the quantum discord. For a bipartite system in the state σ A-cl , the subsystem A is in one of the orthogonal states |α i with probability q i , hence A behaves as a classical system. For this reason, we will call A-classical states the zero-discord states of the form (1). In the literature they are often referred to as the "classical-quantum" states. We denote by C A the set of all A-classical states. By using the spectral decompositions of the σ B|i , any A-classical state σ A-cl ∈ C A can be decomposed as where, for any fixed i, {|β j|i } n B j=1 is an orthonormal basis of H B , and q ij ≥ 0, ij q ij = 1 (note that the |β j|i need not be orthogonal for distinct i's). One defines similarly the set C B of B-classical states, which are the states with zero quantum discord when the subsystem B is measured. A state which is both A-and B-classical possesses an eigenbasis of product vectors. It is fully classical, in the sense that a quantum system in this state can be "simulated" by a classical apparatus being in the state (i, j) with probability q ij .  (2) the pure A-classical (and, similarly, the pure B-classical) states are product states.

B. Distance measures of quantum correlations with the Bures distance
The geometric quantum discord (GQD) of a state ρ of AB has been defined in [14] as the square distance of ρ to the set C A of A-classical states, Instead of taking this distance, we use in this article the Bures distance where ρ and σ are two density matrices and F (ρ, σ) is their fidelity [1,18,41], It is known that (4) defines a Riemannian distance on the convex cone E ⊂ Mat(C, N) of all density matrices of AB. Its metric is equal to the Fubini-Study metric for pure states and coincides (apart from a numerical factor) with the quantum Fisher information which plays an important role in quantum metrology [21]. Moreover, d B satisfies the following properties [1,41]: for any ρ, σ, ρ 1 , ρ 2 , σ 1 , and σ 2 ∈ E, (i) joint convexity of the square distance: if η 1 , η 2 ≥ 0 and η 1 + η 2 = 1, then d B (η 1 ρ 1 + (ii) d B is monotonous under the action of completely positive trace-preserving maps T from . Note that the Hilbert-Schmidt distance d 2 is also unitary invariant but fails to satisfy (ii) (a simple counter-example can be found in [42]). The monotonous Riemannian distances on E have been classified by Petz [20]. The Bures distance can be used to bound from below and above the trace distance d 1 (ρ, σ) = tr(|ρ − σ|) as follows [1], For good reviews on the Uhlmann fidelity and Bures distance, see the book of Nielsen and Chuang [1] and the nice introduction of the article [43] devoted to the estimation of the Bures volume of E.
We define the GQD as The unitary invariance of d B and d 2 implies that D A and D (2) A are invariant under conjugations by local unitaries, By property (ii), D A is monotonous under local operations involving von Neumann measurements on A and generalized measurements on B.
By analogy with (7), one can define two other geometrical measures of quantum correlations: the square distance to the set of classical states C and the geometric measure of entanglement, where F C (ρ) is the maximal fidelity between ρ and a classical state σ cl ∈ C and F S (ρ) the maximal fidelity between ρ and a separable state σ sep ∈ S. The first measure D is a geometrical analogue of the measurement-induced disturbance (MID) [44], which has up to our knowledge not been studied so far (however, an analogue of the MID based on the relative entropy has been introduced in [27]). The second measure E satisfies all criteria of an entanglement measure [17] (in particular, it is monotonous under local operations and classical communication by the property (ii)) and has been studied in [17,22,25]. It is closely related to other entanglement measures [23,24] defined via a convex roof construction thanks to the identity [22] F S (ρ) = max where the maximum is over all The measure E is a geometrical analogue of the entanglement of formation [45]. The latter is defined via a convex roof construction from the von Neumann entropy of the reduced state, Since C ⊂ C A ⊂ S, the three distances are ordered as This ordering of quantum correlations is a nice feature of the geometric measures. In contrast, the entanglement of formation E EoF (ρ) can be larger or smaller than the quantum discord δ A (ρ) [46,47].

III. THE BURES GEOMETRIC QUANTUM DISCORD OF PURE STATES
We first restrict our attention to pure states, for which one can obtain a simple formula for D A in terms of the Schmidt coefficients µ i . We recall that any pure state |Ψ ∈ H A ⊗ H B admits a Schmidt decomposition (12) is unique, the µ i and |ϕ i (respectively |χ j ) being the eigenvalues and eigenvectors of the reduced state (|Ψ Ψ|) A (respectively (|Ψ Ψ|) B ). Note that µ i ≥ 0 and i µ i = Ψ 2 = 1.
where µ max is the largest Schmidt eigenvalue µ i . If this maximal eigenvalue is nondegenerate, the closest A-classical (respectively classical, separable) state to ρ Ψ is the pure product state σ = |ϕ max ⊗ χ max ϕ max ⊗ χ max |, where |ϕ max and |χ max are the eigenvectors corresponding to µ max in the decomposition (12). If µ max is r-fold degenerate, say . . , µ n , then infinitely many A-classical (respectively classical, separable) states σ minimize the distance d B (ρ Ψ , σ). These closest states σ are convex combinations of the orthogonal pure product states |α l ⊗ β l α l ⊗ β l |, l = 1, . . . , r, with is an arbitrary r × r unitary matrix and |ϕ i and |χ i are some eigenvectors in the decomposition (12).
The expression (13) of the geometric measure of entanglement E(ρ Ψ ) is basically known in the literature [23,24]. The closest separable states to pure and mixed states have been investigated in [22]. By inspection of (12) and (13), D A (ρ Ψ ) = 0 if and only if |Ψ is a product state, in agreement with the fact that A-classical pure states are product states (the same holds for the other quantum correlation measures D and E). Moreover, from the The maximal value of D A is reached when µ i = 1/n for any i, that is, for the maximally entangled states (recall that such states are the pure states with reduced states (ρ Ψ ) A and Note that when µ max is r-fold degenerate, the r vectors |α l (respectively |β l ) are or- with eigenvalue µ max . One then obtains another Schmidt decomposition of |Ψ by replacing in (12) the r eigenvectors |ϕ i and |χ i with eigenvalue µ max by |α l and |β l .
Remarkably, the maximally entangled states are the pure states admitting the largest family of closest separable states (this family is a (n 2 + n − 2) real-parameter submanifold of E). For instance, in the case of two qubits (i.e. for n A = n B = n = 2), the Bell states |Φ ± = (|00 ± |11 )/ √ 2 admit as closest separable states the classical states with u * 0l u 0m + u * 1l u 1m = δ ml and q l ≥ 0, q 0 + q 1 = 1. Interestingly, typical decoherence processes such as pure phase dephasing transform ρ Φ ± into one of its closest separable state (|00 00|+|11 11|)/2 at times t ≫ t dec , where t dec is the decoherence time. Slower relaxation processes modifying the populations in the states |00 and |11 do not further increase the distance to the initial state ρ Φ ± . The situation is different for a partially entangled state |Ψ = √ µ 0 |00 + √ µ 1 |11 with µ 1 > µ 0 : then the closest separable state is the pure state |11 , but |Ψ evolves asymptotically to a statistical mixture of |00 and |11 when the qubits are coupled e.g. to thermal baths at positive temperatures.
Replacing σ A-cl in (7) by the right-hand side of (2) we get where we have used ij q ij = 1. Thanks to the Cauchy-Schwarz inequality, for any normal-ized vectors |α ∈ H A and |β ∈ H B one has Let us first assume that µ 1 = µ max > µ 2 , . . . , µ n . Then | α ⊗ β|Ψ | = √ µ max if and only if |α = |ϕ 1 and |β = |χ 1 up to irrelevant phase factors. Thus the maximal fidelity F A (ρ Ψ ) between ρ Ψ and an A-classical state is simply given by the largest Schmidt eigenvalue µ max .
Finally, the last inequality in (17) is an equality if and only if both sums inside the square brackets are equal to unity, i.e. |α ∈ span{|ϕ i } n i=1 and |β ∈ span{|χ i } n i=1 (this holds trivially if n A = n B = n). Putting all conditions together, we obtain |α ∈ V max , |β ∈ W max , and β|χ i = e iθ ϕ i |α for i = 1, . . . , r. Therefore, from any orthonormal family {|α l } r l=1 of V max one can construct r orthogonal vectors |α l ⊗ β l satisfying | α l ⊗ β l |Ψ | = √ µ max for all l = 1, . . . , r, with β l |χ i = ϕ i |α l . The probabilities {q ij } maximizing the sum inside the brackets in (15) are given by q ij = q i if i = j ≤ r and zero otherwise, where {q l } r l=1 is an arbitrary set of probabilities. The corresponding A-classical states with maximal fidelities F (ρ Ψ , σ) are the classical states σ = r l=1 q l |α l ⊗ β l α l ⊗ β l |.
The equality between the correlation measures D A , D, and E is a consequence of the fact that the closest states to ρ Ψ are classical states. Such an equality is reminiscent from the equality between the entanglement of formation E EoF and the quantum discord δ A for pure states. Let us notice that it does not hold for the Hilbert-Schmidt distance, for which the closest A-classical state to a pure state is in general a mixed state. Actually, one infers from the expression that the closest A-classical state results from a competition between the maximization of the fidelity F (ρ Ψ , σ A-cl ) and the minimization of the trace tr(σ 2 A-cl ), which is maximum for pure states. For instance, one can show [39] that the closest A-classical states to the Bell states |Φ ± for d 2 are mixed two-qubit states. The validity of Theorem 1 is one of the major advantage of the Bures-GQD over the Hilbert-Schmidt-GQD.

A. Link with minimal error quantum state discrimination
The determination of D A (ρ) is much more involved for mixed states than for pure states.
We show in this section that this problem is related to ambiguous QSD. As it has been recalled in the introduction, in ambiguous QSD a state ρ i drawn from a known family {ρ i } n A i=1 with prior probabilities {η i } n A i=1 is sent to a receiver. The task of the latter is to determine which state he has received with a maximal probability of success. To do so, he performs a generalized measurement and concludes that the state is ρ j when his measurement result is j. The generalized measurement is given by a family of positive operators M i ≥ 0 satisfying i M i = 1 (POVM). The probability to find the result j is P j|i = tr(M j ρ i ) if the system is in the state ρ i . The maximal success probability of the receiver reads Theorem 2. Let ρ be a state of the bipartite system AB with Hilbert space H = H A ⊗ H B and let α = {|α i } n A i=1 be a fixed orthonormal basis of H A . Consider the subset C A (α) ⊂ C A of all A-classical states σ A-cl such that α is an eigenbasis of tr B (σ A-cl ) (i.e. C A (α) is the set of all states σ A-cl of the form (1), for arbitrary probabilities q i and states σ B|i on H B ). Then the maximal fidelity F (ρ, C A (α)) = max σ A-cl ∈C A (α) F (ρ, σ A-cl ) of ρ to this subset is equal to where P opt v.N.
is the maximal success probability over all von Neumann measurements given by orthogonal projectors Π i of rank n B (that is, self-adjoint operators on H satisfying Π i Π j = δ ij Π i and dim(Π i H) = n B ), and (if η i = 0 then ρ i is not defined but does not contribute to the sum in (20)).
This theorem will be proven in Sec. IV B. Note that the ρ i are quantum states of AB if η i > 0, because the right-hand side of the last identity in (21) is a non-negative operator and η i is chosen such that tr(ρ i ) = 1. Moreover, {η i } n A i=1 is a set of probabilities (since η i ≥ 0 and i η i = tr(ρ) = 1) and {ρ i , η i } n A i=1 defines a convex decomposition of ρ, i.e. ρ = i η i ρ i . Let us assume that ρ is invertible. Then the application of a result by Eldar [48] shows that the POVM maximizing the success probability P S ({ρ i , η i }) in (19) is a von Neumann measurement with projectors Π i of rank n B , i.e. F (ρ, C A (α)) = P opt v.N.
In fact, one may first notice that all matrices ρ i have rank r i = n B (for indeed, ρ i has the same rank as η i ρ −1/2 ρ i = |α i α i | ⊗ 1 √ ρ and the latter matrix has rank n B ). Next, we argue that the ρ i are linearly independent, in the sense that their eigenvectors |ξ ij form a linearly independent family {|ξ ij } j=1,...,n B i=1,...,n A of vectors in H. Actually, a necessary and sufficient condition for |ξ ij to be an eigenvector of ρ i with eigenvalue λ ij > 0 is |ξ ij = (λ ij η i ) −1 √ ρ|α i ⊗ |ζ ij , |ζ ij ∈ H B being an eigenvector of R i = α i |ρ|α i with eigenvalue λ ij η i > 0. For any i, the Hermitian invertible matrix R i admits an orthonormal eigenbasis j=1 . Thanks to the invertibility of √ ρ, {|ξ ij } j=1,...,n B i=1,...,n A is a basis of H and thus the states ρ i are linearly independent. It is shown in [48] that for such a family of linearly independent states the second equality in (22) holds true.
The following result on the Bures-GQD of mixed states is a direct consequence of Theorem 2.
Theorem 3. For any state ρ of the bipartite system AB, the fidelity to the closest A-classical state is given by where the maxima are over all orthonormal basis {|α i } of H A and all orthogonal families of projectors of H A ⊗ H B with rank n B . Hence, using the notation of theorem 2, If ρ > 0 then one can replace P opt v.N. It is noteworthy to observe that the basis vectors |α i can be recovered from the states ρ i and probabilities η i by forming the square-root measurement operators M i = η i ρ −1/2 ρ i ρ −1/2 , with ρ = i η i ρ i (we assume here ρ > 0). Actually, such measurement operators are equal to the rank-n B projectors M i = |α i α i | ⊗ 1. By bounding from below P opt v.N.
the success probability corresponding to Π i = M i , we obtain The square root measurement plays an important role in the discrimination of almost orthogonal states [49,50] and of ensembles of states with certain symmetries [32,33].
To illustrate our result, let us study the ambiguous QSD task for some specific states ρ.  (21) entails if ρ is invertible. Thus ρ is A-classical (this was of course to be expected since D A (ρ) = 0 if and only if ρ is A-classical, see Sec. II). Therefore, we can interpret our result (24) as follows: the non-zero discord states ρ are such that the states (21) are non-orthogonal and thus cannot be perfectly discriminated for any orthonormal basis {|α i } of H A .
(ii) If ρ = ρ Ψ is a pure state, then all ρ i with η i > 0 are identical and equal to ρ Ψ , so that Proposition. If n A ≤ n B , the smallest fidelity F A (ρ) for all states ρ of AB is equal to 1/n A .
If rn A ≤ n B < (r + 1)n A with r = 1, 2, . . ., the states ρ with F A (ρ) = 1/n A are any convex combinations of the r maximally entangled pure states |Ψ k = n We deduce from this result that the GQD D A (ρ) varies between 0 and 2 − 2/ √ n A when n A ≤ n B . By virtue of theorem 1, the proposition, and the inequality E(ρ) ≤ D A (ρ), the geometric measure of entanglement E(ρ) also varies between these two values. This means that the most distant states from the set of A-classical states C A are also the most distant from the set of separable states S. If n A ≤ n B < 2n A , these most distant states are always maximally entangled pure states.
Proof. The success probability P opt v.N.
S must be clearly larger than the highest prior probability η max = max i {η i }. (A receiver would obtain P S = η max by simply guessing that his state is ρ imax , with η imax = η max ; a better strategy is of course to perform the von Neumann measurement {Π i } such that Π imax projects on a n B -dimensional subspace containing the range of ρ imax ; this range has a dimension ≤ n B by a similar argument as in the discussion following theorem 2.) In view of (24) and by using η max ≥ 1/n A (since i η i = 1) we get for any mixed state ρ.
When n A ≤ n B the bound (26) is optimum, the value 1/n A being reached for the maximally entangled pure states, see Sec. III. Thus 1/n A is the smallest possible fidelity. Let ρ be a state having such a fidelity F A (ρ) = 1/n A . According to (24) and since it has been argued before that P opt v.N.

S
({ρ i , η i }) = 1/n A whatever the orthonormal basis {|α i }. It is intuitively clear that this can only happen if the receiver gets a collection of identical states ρ i with equal prior probabilities η i = 1/n A .
A rigorous proof of this fact is given in the appendix. From (21) and ρ = η i ρ i we then obtain α i |ρ A |α i = 1/n A and ρ i = ρ for any i = 1, . . . , n A and any orthonormal basis The first equality implies ρ A = 1/n A . By replacing the spectral decomposition (21), the second equality yields tr B (|Ψ k Ψ l |) = n −1 A δ kl for all k, l with p k p l = 0. Taking advantage of this identity for k = l, one finds that the eigenvectors |Ψ k of ρ with positive eigenvalues p k have all their Schmidt eigenvalues equal to 1/n A , that is, their Schmidt decompositions read |Ψ k = n If n B < (r + 1)n A then at most r subspaces V (k) B may be pairwise orthogonal. Thus at most r eigenvalues p k are non-zero.
Let us now discuss the case n A > n B . In that case the smallest value of the maximal fidelity F S (ρ) to a separable state is equal to 1/n B and F S (ρ) = 1/n B when ρ is a pure maximally entangled state. This is a consequence of (9) and of the bound F S (ρ Ψ ) ≥ 1/n B for pure states ρ Ψ (see Sec. III). As a result, the geometric measure of entanglement E(ρ) varies between 0 and 2 − 2/ √ n with n = min{n A , n B }, in both cases n A ≤ n B and n B > n A .
We could not establish a similar result for the GQD D A (ρ). When n A > n B , the bound (26) is still correct but it is not optimal, i.e. there are no states ρ with fidelities F A (ρ) equal to 1/n A . Indeed, following the same lines as in the proof above, one shows that if F A (ρ) = 1/n A then the eigenvectors |Ψ k of ρ with non-zero eigenvalues must have maximally mixed marginals [ρ Ψ k ] A = 1/n A . But this is impossible since rank([ρ Ψ k ] A ) ≤ n B by (12).
According to the results of Sect. III, pure states ρ Ψ have fidelities F A (ρ Ψ ) ≥ 1/n B , so one may expect that states close enough to pure states have fidelities close to 1/n B or larger.
This can be shown rigorously by invoking the bound where ρ is the norm of ρ and δ ρ = 0 if rank(ρ) ≤ n B and 1 otherwise. This bound can be established as follows. Let us write ρ = p|Ψ Ψ| + (1 − p)ρ ′ where |Ψ is the eigenvector of ρ with maximal eigenvalue p = ρ and the density matrix ρ ′ has support on [C|Ψ ] ⊥ .
Choosing an orthonormal family {Π i } of projectors of rank n B satisfying Π 1 |Ψ 1 = |Ψ 1 , we get from (24) Let us fix the orthonormal basis {|α i } such that |α 1 is the eigenvector with maximal eigenvalue µ max in the Schmidt decomposition (12) of |Ψ . This leads to the maximal possible value pµ max of the first term in the right-hand side of (28). We now bound the sum in this right-hand side by its i m th term Collecting the above results and using the inequalities µ max ≥ 1/n B and η ′ max ≥ 1/n A (since n A i=1 η ′ i = 1), one gets (27). Note that this bound is stronger than (26) only for states ρ satisfying ρ ≥ (1 + n A − n B ) −1 or rank(ρ) ≤ n B . In summary, we can only conclude from the analysis above that when n A > n B the smallest possible fidelity min ρ∈E F A (ρ) lies in the interval (1/n A , 1/n B ]. B. Derivation of the variational formula (23) To prove theorems 2 and 3, we start by evaluating the trace norm in (5) by means of the formula T 1 = max U | tr(UT )|, the maximum being over all unitaries on H. By (2), In the third line we have replaced the maximum over unitaries U by a maximum over all orthonormal basis {|Φ ij } of H (with |Φ ij = U † |α i ⊗ β j|i ). The last equality in (29) can be explained as follows. The expression in the last line is clearly larger than that of the third line; since for any i and j one can choose the phase factors of the vectors |Φ ij in such a way that Φ ij | √ ρ|α i ⊗ β j|i ≥ 0, the two expressions are in fact equal.
One has to maximize the last member of (29) over all families of i-dependent orthonormal basis {|β j|i } of H B and all probabilities q ij . The maximum over the probabilities q ij is easy to evaluate by using the Cauchy-Schwarz inequality and i,j q ij = 1. It is reached for We thus obtain where we have set |ψ j|i = α i | √ ρ|Φ ij ∈ H B . We proceed to evaluate the maximum over By the Cauchy-Schwarz inequality, for any fixed i one has Note that (33) is an inequality if the vectors |ψ j|i are orthogonal for different j's. We now show that this is the case provided that the |Φ ij are chosen appropriately. In fact, let us take an arbitrary orthonormal basis {|Φ ij } of H and consider the Hermitian n B ×n B matrix S (i) with coefficients given by the scalar products S (i) jk = ψ j|i |ψ k|i . One can find a unitary matrix V (i) such that S (i) = (V (i) ) † S (i) V (i) is diagonal and has non-zero diagonal elements in the first r i raws, where r i is the rank of lj |ψ l|i form an orthogonal set {| ψ j|i } r i j=1 and vanish for j > r i . Therefore, for any fixed orthogonal family {Π i } n A i=1 of projectors of rank n B , there exists an orthonormal basis {|Φ ij } of H such that (32) holds and the inequality in (33) is an equality. Substituting this equality into (31), one finds which yields the result (20). The formula (23) is obtained by maximization over the basis

C. Closest A-classical states
The proof of the previous subsection also gives an algorithm to find the closest A-classical states to a given mixed state ρ. To this end, one must find the orthonormal basis {|α opt (24) (23)).
Proof. This follows directly from the proof of the previous subsection. Actually, by using the expression (30) of the optimal probabilities q ij and the fact that the Cauchy-Schwarz inequality (33) is an equality if and only if |β j|i = |ψ j|i / ψ j|i when ψ j|i = 0, we conclude that the closest A-classical states to ρ are given by (1) with |α i = |α opt i and The denominator is equal to F A (ρ), see (34). The numerator is the same as the second factor in the right-hand side of (35). For indeed, by construction | ψ j|i = α opt i | √ ρ| Φ ij and Let us stress that the optimal measurement {Π opt i } and basis {|α opt i } may not be unique, so that ρ may have several closest A-classical states σ ρ . This is the case for instance when ρ is a pure state with a degenerate maximal Schmidt eigenvalue, as we have seen in theorem 1. If B|i are two closest A-classical states to ρ with the same basis {|α opt i } then so are all convex combinations σ ρ (η) = ησ ρ + (1 −η)σ ′ ρ with 0 ≤ η ≤ 1. This fact is a direct consequence of the convexity of the square Bures distance (property (i) in Section II), given that σ ρ (η) ∈ C A . As a result, states ρ having more than one closest A-classical state will generally admit a continuous family of such states.

V. CONCLUSIONS
We have established in this paper a link between ambiguous QSD and the problem of finding the minimal Bures distance of a state ρ of a bipartite system AB to a state with vanishing quantum discord. More precisely, the maximal fidelity between ρ and an A-classical (i.e. zero-discord) state coincides with the maximal success probability in discriminating the As stated in the introduction, the QSD task can be solved for n A = 2 states. Thus the aforementioned results provide a method to find the geometric discord D A and the closest A-classical states for bipartite systems composed of a qubit A and a subsystem B with arbitrary space dimension n B ≥ 2. In particular, explicit formulae can be derived for two qubits in states with maximally mixed marginals and for (n B + 1)-qubits in the DQC1 algorithm [39]. For subsystems A with higher space dimensions n A > 2, several open issues deserve further studies. Firstly, it would be desirable to characterize the "most quantum" states when n A > n B . Secondly, it is not excluded that the specific QSD task associated to the minimal Bures distance admits an explicit solution. Thirdly, the relation of D A with the geometric measure of entanglement in tripartite systems should be investigated; in particular, there may exist some inequality analogous to the monogamy relation [15] between the quantum discord and the entanglement of formation.
Let us emphasize that our results may shed new light on dissipative dynamical processes involving decoherence, i.e. evolutions towards classical states. In fact, our analysis may allow in some cases to determine the geodesic segment linking a given state ρ 0 with non-zero discord to its closest A-classical state σ ρ 0 . Such a piece of geodesic is contained in the set of all states ρ having the same closest A-classical state σ ρ = σ ρ 0 as ρ 0 . It would be of interest to compare in specific physical examples the Bures geodesics with the actual paths followed by the density matrix during the dynamical evolution.
Acknowledgements: We are grateful to Maria de los Angeles Gallego for revising the derivation of Eq.(23) and for providing us some notes about the calculation of the geometric discord for Werner states. D.S. thanks Aldo Delgado for interesting discussions. We acknowledge financial support from the French project no. ANR-09-BLAN-0098-01, the Chilean Fondecyt project no. 100039, and the project Conicyt-PIA anillo no. ACT-1112 "Red de Análisis estocástico y aplicaciones".
Appendix A: Necessary and sufficient condition for the optimal success probability to be equal to the inverse number of states Let {ρ i } n A i=1 be a family of n A states on H with prior probabilities η i , where n A = N/n B is a divisor of dim(H) = N. We assume that the ρ i have ranks rank(ρ i ) ≤ n B for any i. Let P opt v.N. S ({ρ i , η i }) be the optimal success probability in discriminating the states ρ i , defined by Eq. (20). We prove in this appendix that P opt v.N. The conditions η i = 1/n A and ρ i = ρ are clearly sufficient to have P opt v.N.

S
({ρ i , η i }) = 1/n A (a measurement cannot distinguish the identical states ρ i and thus cannot do better than a random choice with equal probabilities). We need to show that they are also necessary conditions. Let us assume P opt v.N. S ({ρ i , η i }) = 1/n A . The equality η i = 1/n A for all i is obvious from the bounds P opt v.N. S ({ρ i , η i }) ≥ η max ≡ max i {η i } and η max ≥ 1/n A (see Sec. IV A). Therefore, according to our hypothesis, any orthogonal family {Π i } n B i=1 of projectors of rank n B satisfies i tr(Π i ρ i ) = n A P S ({ρ i , η i }) ≤ 1. We now argue that the states ρ i have ranges contained in a common subspace V. In fact, let V be the n B -dimensional subspace of H spanned by the eigenvectors of ρ 1 associated to the n B highest eigenvalues (including degeneracies), and let us denote by Π 1 the projector onto V. Then ρ 1 H ⊂ V (since we have assumed rank(ρ 1 ) ≤ n B ) and thus ρ 1 = Π 1 ρ 1 . Thanks to the inequality above, 1 ≥ i tr(Π i ρ i ) ≥ tr(Π 1 ρ 1 ) = 1. It follows that tr(Π 2 ρ 2 ) = 0 for any projector Π 2 of rank n B orthogonal to Π 1 . Hence ρ 2 , and similarly all ρ i , i = 3, . . . , n A , have ranges contained in V. This proves the aforementioned claim.
In order to show that all the states ρ i are identical, we further introduce, for each 1 ≤ k ≤ n B , some n B -dimensional subspace V (k) containing the eigenvectors associated to the k highest eigenvalues λ 1 ≥ . . . ≥ λ k of ρ 1 , the other eigenvectors being orthogonal to V (k) (then V (n B ) = V). We also choose a n B -dimensional subspace W (k) ⊂ H orthogonal to V (k) be an orthogonal family of projectors of rank n B such that Π 1 ρ 2 ) is smaller than the sum of the k highest eigenvalues of ρ 2 (including degeneracies). By (A1), this sum is larger than the sum λ 1 + · · · + λ k of the k highest eigenvalues of ρ 1 . By exchanging the roles of ρ 1 and ρ 2 , we obtain the reverse equality. Since moreover k is arbitrary between 1 and n B , it follows that ρ 1 and ρ 2 have identical eigenvalues. By using (A1) again, tr(Π (k) 1 ρ 2 ) is equal to the sum of the k highest eigenvalues of ρ 2 . Hence the k corresponding eigenvectors of ρ 2 are contained in the kdimensional subspace V (k) ∩ V. Since k is arbitrary, this proves that ρ 1 and ρ 2 have identical eigenspaces. Therefore ρ 1 = ρ 2 . Repeating the same argument for the other states ρ i , i ≥ 3, we obtain ρ 1 = · · · = ρ n A .