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Geometric quantum discord with Bures distance: the qubit case

The minimal Bures distance of a quantum state of a bipartite system AB to the set of classical states for subsystem A defines a geometric measure of quantum discord. When A is a qubit, we show that this geometric quantum discord is given in terms of the eigenvalues of a (2 n_B) x (2 n_B) hermitian matrix, n_B being the Hilbert space dimension of the other subsystem B. As a first application, we calculate the geometric discord for the output state of the DQC1 algorithm. We find that it takes its highest value when the unitary matrix from which the algorithm computes the trace has its eigenvalues uniformly distributed on the unit circle modulo a symmetry with respect to the origin. As a second application, we derive an explicit formula for the geometric discord of two-qubit states with maximally mixed marginals and compare it with other measures of quantum correlations. We also determine the closest classical states to such two-qubit states.


I. INTRODUCTION
In order to understand the origin of quantum speedups in quantum algorithms and to analyze the specificities of quantum communication protocoles, it is of prime importance to identify the resources needed to have a quantum advantage [1]. Despite substantial progress in the last decades [2,3], a complete characterization of quantum correlations (QCs) in composite quantum systems has not yet emerged, even for bipartite systems. Furthermore, the role played by these correlations in information processing tasks remains to a large extent unclear. The quantum discord (QD) is one measure of QCs in bipartite systems [4,5]. It coincides with the entanglement of formation [6] for pure states, but for mixed states it quantifies QCs that may be present even in separable states. The discord is believed to be a better indicator than entanglement of the "degree of quantumness" of mixed states. Moreover, it has been suggested [7][8][9][10] that it could capture the relevant quantum resource in Knill and Laflamme's algorithm of deterministic quantum computation with one qubit (DQC1) [11]. This algorithm allows to compute efficiently the trace of a 2 n × 2 n unitary matrix U n . It involves a polarized control qubit, which remains unentangled with n unpolarized target qubits at all stages of the computation. The amount of entanglement for any bipartition of the (n + 1) qubits is bounded independently of n [12]. However, a non-vanishing QD between the control and the n target qubits appears in the output state [7][8][9][10], save for some particular unitaries U n [8,13]. * Dominique.Spehner@ujf-grenoble.fr In a recent article [14], we have proposed to use the Bures distance to the set of zero-discord states as a geometrical analogue of the QD. This geometric QD (GQD) does not suffer the drawbacks of a similar measure introduced by Dakić, Vedral, and Brukner [13], which makes use of the Hilbert-Schmidt distance. In particular, for pure states the Bures-GQD is equal to the geometric measure of entanglement defined as the Bures distance to the set of separable states [15,16]. The main advantage of the geometric approach is that, in addition to quantifying the degree of quantumness of a given state ρ, one can look for the closest zero-discord states to ρ. This may help to understand decoherence processes and peculiar features of QCs during dynamical evolutions, such as the sudden transitions discussed in [17,18]. The main result of Ref. [14] is that the Bures-GQD of a mixed state ρ is equal to the maximal success probability in ambiguous quantum state discrimination of a family of states ρ i depending on ρ. Moreover, the closest zero-discord states to ρ are given in terms of the optimal von Neumann measurement in this discrimination task.
In this paper, we apply the aforementioned results [14] in order to calculate the Bures-GQD when one subsystem of the bipartite system is a qubit. In particular, we determine the GQD between the control qubit and the n target qubits of the output state in the DQC1 algorithm. We show that for any unitary matrix U n , the discord is bounded from above by its value obtained by choosing random unitaries distributed according to the Haar measure. We also derive an explicit formula for states ρ of two qubits with maximally mixed marginals and determine the closest zero-discord states to ρ. We compare our results to those obtained by using the Hilbert-Schmidt distance and the relative entropy.
The layout of the paper is as follows. We recall in Sec. II the problem of ambiguous quantum state discrimination (QSD), the definitions of the quantum discords, and the link between the geometric discord and QSD. In Sec. III, we show how to solve the QSD task and determine the GQD when the measured subsystem is a qubit. We calculate in Sec. IV the GQD for the output state in the DQC1 model. Sec. V is devoted to the case of two qubits. We compare our results for the GQD with the other quantum discords and find the closest zero-discord states to a Bell-diagonal state. The last section VI contains conclusive remarks. Some technical details are presented in the appendix.

II. QUANTUM STATE DISCRIMINATION AND THE GEOMETRIC QUANTUM DISCORD
A. Ambiguous quantum state discrimination The objective of quantum state discrimination (QSD) consists in distinguishing states taken randomly from a known ensemble of states [19,20]. If these states are non-orthogonal, any measurement devised to distinguish them cannot succeed to identify exactly which state from the ensemble has been chosen. The QSD task is to find the optimal measurements leading to the smallest probability of equivocation. More precisely, a receiver is given a state ρ i in a finite dimensional Hilbert space H, drawn from a known family {ρ i } nA i=1 , with a prior probability η i . In order to determine which state he has received, he performs a generalized measurement and concludes that the state is ρ j when he gets the measurement outcome j. The probability to find this outcome given that the state is ρ i is P j|i = tr(M j ρ i ), where {M j } is a family of nonnegative measurement operators satisfying j M j = 1 (POVM). In the so-called ambiguous QSD strategy, the number of measurement outcomes is chosen to be equal to the number n A of states in the ensemble {ρ i , η i }. The maximal success probability of the receiver reads If the ρ i have rank n B = dim(H)/n A and are linearly independent, in the sense that their eigenvectors |ξ ij with nonzero eigenvalues form a linearly independent family {|ξ ij } j=1,...,nB i=1,...,nA of vectors in H, it is known [21] that the optimal POVM {M opt i } nA i=1 is a von Neumann measurement with projectors of rank n B . Therefore, in that case is the maximum success probability over all orthogonal families {Π i } nA i=1 of projectors of rank n B (i.e., self-adjoint operators satisfying Π i Π j = δ ij Π i and rank(Π i ) = n B ).

B. Quantum discords
We briefly recall the definition of the QD of Ollivier and Zurek and Henderson and Vedral [4,5]. Let AB be a bipartite system in the state ρ. The mutual information of AB is given by where S(·) stands for the von Neumann entropy and ρ A = tr B (ρ) and ρ B = tr A (ρ) are the reduced states of A and B, respectively. The mutual information is nonnegative by the sub-additivity of S. It characterizes the total (classical and quantum) correlations in AB. The QD measures the amount of mutual information which is not accessible by local measurements on the subsystem A. It can be defined as where the maximum is over all von Neumann measurements (i.e., orthogonal families of projectors) is the postmeasurement state in the absence of readout. The second term in (3) represents the amount of classical correlations. It can be shown that δ(ρ) ≥ 0 and δ A (σ A-cl ) = 0 if and only if where {|α i } nA i=1 is an orthonormal basis for subsystem A, σ B|i are some (arbitrary) states of B depending on the index i, and q i ≥ 0 are some probabilities. We call Aclassical states the zero-discord states of the form (4) [22].
The set of quantum states can be equipped with various distances. From a quantum information perspective, it is natural to study the geometry induced by the Bures distance [1,23,24] where F (ρ, σ) is the Uhlmann fidelity generalizing the usual pure-state fidelity | Ψ|Φ | 2 . The distance d B pertains to the family of monotonous (that is, contractive with respect to completely positive tracepreserving maps) Riemannian distances [25]. Its metric coincides with the quantum Fisher information playing an important role in quantum metrology [26]. We take this opportunity to mention a misprint in our previous article [14]: although the square distance d B (ρ, σ) 2 is jointly convex, this is not the case for d B (ρ, σ). The geometric QD (GQD) is by definition the square distance of ρ to the set C A of A-classical states, One can similarly define the discord D B (ρ) = d B (ρ, C B ) 2 , where C B is the set of B-classical states. One has in general D B = D A (in fact, it is clear from the form (4) of the A-classical states that C B = C A ). The same holds for the discords δ A = δ B , because the maximal mutual informations after measuring subsystems A or B are in general different. It is not difficult to show [14] that for pure states D A and D B coincide with the geometric measure of entanglement E(ρ) = min σsep∈S d B (ρ, σ sep ) 2 , where S denotes the convex set of separable states. More , µ max being the highest eigenvalue of (ρ Ψ ) A (maximal Schmidt coefficient). This equality between D A , D B , and E comes from the fact that the closest separable state to a pure state is a pure product state [27].

C. Link between DA and state discrimination
The evaluation of the GQD (7) for mixed states ρ turns out to be related to an ambiguous QSD task [14]. More precisely, the fidelity between ρ and its closest A-classical state is given by the maximum success probability (2), In the right-hand side, the maximum is over all orthonormal basis is an ensemble of states depending on {|α i } and ρ defined by (if η i = 0 then ρ i is not defined but does not contribute to the success probability). The number of states ρ i is equal to the dimension n A of the Hilbert space of A. It is easy to see that if ρ > 0 then all the ρ i have ranks equal to the dimension n B of the space of B. Furthermore, they are linearly independent. Thus P opt v.N.

S
can be replaced by P opt S in Eq. (8). Note that {ρ i , η i } defines a convex decomposition of ρ, ρ = i η i ρ i . A remarkable property of this decomposition is that the associated square-root measurement operators M i = η i ρ −1/2 ρ i ρ −1/2 , which are known to be optimum in the ambiguous QSD of symmetric ensembles [28,29], coincide with the projectors Π i = |α i α i | ⊗ 1. By bounding from below P opt v.N.
by the success probability corresponding to Π i , we obtain Let us denote by {|α opt i } and {Π opt i } the basis and projective measurement(s) maximizing P opt v.N. (8). Then the closest A-classical state(s) to ρ is (are) [14] Finding analytically the optimal success probability and optimal measurement(s) in ambiguous QSD with n A > 2 states is an open problem, excepted for symmetric ensembles [20,28,29]. A necessary and sufficient condition for a POVM to be optimum is due to Helstrom [19]. Various bounds on P opt S have been derived in the literature (see [30] and references therein). Moreover, efficient methods are available to solve this problem numerically [31,32]. We focus in the next section on the simpler case n A = 2, for which an analytic solution is well-known.
If the subsystem A is a qubit, the ensemble {ρ i , η i } contains only n A = 2 states and the QSD task can be easily handled [19,20]. In our case, we must maximize the success probability over all von Neumann measurements given by orthogonal projectors Π 0 and Π 1 having the same rank n B . One starts by writing the projector Π 1 as 1 − Π 0 in the expression of the success probability, with Λ = η 0 ρ 0 − η 1 ρ 1 . Thanks to the min-max principle [33], the maximum of tr(Π 0 Λ) over all projectors Π 0 of rank n B is equal to the sum of the n B highest eigenvalues λ 1 ≥ · · · ≥ λ nB of the hermitian matrix Λ, and the optimal projector Π opt 0 is the spectral projector associated to these highest eigenvalues. One has For the states ρ i and probabilities η i defined by Eq.(9), is an orthonormal basis of C 2 . For any such basis the operator inside the parenthesis in the last formula is equal to σ u ≡ 3 m=1 u m σ m for some unit vector u ∈ R 3 (here σ 1 , σ 2 , and σ 3 are the Pauli matrices). Reciprocally, one can associate to any unit vector u ∈ R 3 the eigenbasis where λ l (u) are the eigenvalues in non-increasing order of the 2n B × 2n B hermitian matrix Note that −ρ ≤ Λ(u) ≤ ρ, so that nB l=1 λ l (u) ≤ nB l=1 p l , p l being the eigenvalues of ρ in non-increasing order.
If ρ > 0 then Λ(u) has n B positive and n B negative eigenvalues. In such a case (13) reduces to the well known expression P opt v.N.
It is shown in [14] that the minimal value that F A (ρ) can take is equal to 1/n A when n A ≤ n B [34]. If A is a qubit, the maximal value of the GQD is thus D A = 2− √ 2, see (5). It is convenient to work with a normalized discord, given in terms of F A (ρ) by

IV. GEOMETRIC DISCORD IN THE DQC1 MODEL
In this section, we determine the Bures-GQD for the output state of the DQC1 algorithm. We obtain an analytic expression of the discord for all unitary matrices U n from which the algorithm computes the trace.
The DQC1 model [11] consists in a control qubit, labelled by the index 0, coupled to n target qubits 1, 2, . . . , n. The control qubit is initially in a mixture of the standard basis vectors |0 and |1 with populations (1 ± α)/2 differing from one half. The target qubits are in the completely mixed state 2 −n 1 n , where 1 n stands for the identity operator on the 2 n -dimensional space H B of qubits 1, . . . , n. This state approximates well the thermal state of the nuclear spins in liquid-state nuclear magnetic resonance (NMR) experiments at room temperature. The initial state of the (n + 1) qubits reads with α ∈ [−1, 1], α = 0. This state is transformed by the following circuit composed of a Hadamard gate H and a unitary gate U n acting on H B controlled by the qubit 0: The output state is with N = 2 n . Hence the reduced state tr 1,...,n (ρ U,α n+1 ) of the qubit 0 contains some information about the normalized trace z n = tr(U n )/N . No efficient classical algorithm to compute z n is known. The DQC1 quantum algorithm provides accurate approximations of σ 1 ⊗1 n out = α Re z n and σ 2 ⊗1 n out = α Im z n after sufficiently many runs of the measurement of the spin σ 1 and σ 2 on the qubit 0. The number of runs is independent of n and scales logarithmically with the error probability. In that sense, the DQC1 algorithm is exponentially more efficient than all known classical algorithms for estimating the normalized trace of U n [12]. Moreover, it works whatever the value of α provided that α = 0, hence demonstrating the "power of even the tiniest fraction of a qubit" [12].
The control qubit is unentangled with the target qubits, as can be shown from the relation where |u k are the eigenvectors of U n with eigenvalues e iω k , |ϕ k = cos δ|0 + e iω k sin δ|1 , and |χ k = sin δ|0 + e iω k cos δ|1 , with sin(2δ) = α. For other bipartitions of the (n + 1) qubits, e.g., putting together in one subsystem the control qubit and half of the target qubits, the entanglement does not vanish in general but it is bounded uniformly in n [12]. For large system sizes, the total amount of bipartite entanglement is thus a negligible fraction of the maximal entanglement possible.
In what follows, we split the (n + 1) qubits into the subsystem A containing the qubit 0 and the subsystem B containing the qubits 1, . . . , n. It is clear from (20) that the output state is B-classical, hence it has a vanishing discord δ B . It was shown in [7] that this state has generically a non-vanishing discord δ A . The term "generically" refers here to a random choice of the unitary U n with the Haar distribution. This presence of a nonzero discord has been demonstrated experimentally in optical [8] and liquid-state NMR [9] implementations of DQC1.
The Bures-GQD of the output state (19) can be easily determined with the help of (14). The eigenvalues of ρ U,α n+1 are (1 ± α)/(2N ) and the two corresponding eigenspaces are spanned by the N vectors |0 j ± |1 U n j , where |j , j ∈ {0, 1} n , are the standard basis vectors of H B . This yields the square root Substituting this expression into (15) and introducing the angles θ, φ such that u = (sin θ cos φ, sin θ sin φ, cos θ), one finds denote the real and imaginary parts of the operator O. By diagonalizing each of the four blocks of the matrix appearing between V and V † in the right-hand side of (22), one sees that its eigenvalues λ k,± (u) are the eigenvalues of k distinct 2 × 2 matrices. This yields One has clearly ±λ k,± (u) ≥ 0. Writing the maximum over u in (14) as a maximum over θ and φ and noting that the maximum over θ is reached for sin 2 θ = 1, we get (24) Let us point out that the maximal fidelity (24) to the A-classical states is a decreasing function of α 2 . Hence the geometric discord D A (ρ U,α n+1 ) increases with the initial purity (1 + α 2 )/2 of the control qubit.
For large system sizes n, the sum over k in (24) can be replaced by an integral over the smooth normalized spectral density n(ω) of the eigenphases ω k of U n , with n S (ω) = n(ω)+n(ω+π). Let us show that the smallest fidelity is achieved for unitary operators U n having a constant symmetrized spectral density n S (ω) = 1/π. Actually, by substituting max φ by 2π 0 dφ/(2π) in (25) and using 2π 0 dφ n S (φ) = 2, one gets the bound The inequality is an equality if and only if the integral in (25) is independent of φ. Expanding n S as a Fourier series, n S (ω) = p a p e 2ipω , this is equivalent to , one easily sees that the integrals in (27) do not vanish for any p. Therefore a p = 0 when p = 0 and n S (ω) is constant. Reciprocally, if n S (ω) is constant then for any α ∈ [−1, 1] the inequality in (26) is an equality.
Let us emphasize that the bound (26) is satisfied for all n ≥ 1 (in fact, it can be obtained by replacing as before the maximum over φ by an integral, but in Eq.(24) instead of (25)). As a consequence, whatever the unitary matrix U n and the system size n, the GQD is always smaller or equal to the GQD for an infinite unitary matrix U unif with constant symmetrized spectral density n S . Such matrices have equidistributed eigenvalues on the unit circle modulo a symmetry with respect to the origin. This highest possible discord is achieved in particular for random unitaries distributed according to the Haar measure on the unitary group U (N ), since then n(ω) = 1/(2π) almost surely in the large n limit [7].
The normalized GQD (16) of the output state (19) is shown in Fig. 1 as a function of α for the optimal unitary U unif with constant n S (ω) and for the rotation operator where 3 /2 is the total angular momentum of the n spins in the z direction. In the second case, the fidelity is determined from (24) by using the nonperiodic spectral density n(ω) = (2/π) 3/2 exp(−8 ω 2 /π 2 ) and noting that the maximum is reached for φ = 0. This Gaussian density is obtained by using the fact that the eigenphases of U n , ω k = π(k − n/2)/(2 √ n) with k = 0, . . . , n, have multiplicities d k = n! k!(n−k)! satisfying 2 −n d k ∼ 2 πn exp[−2(k − n 2 ) 2 /n] in the limit n → ∞ [note that the scaling like 1/ √ n of the rotation angle in (28) is dictated by the wish to obtain a nontrivial n(ω)]. We also compare in Fig. 1 the geometric discord with the quantum discord (3), using the result of Ref. [7] apply- ing to random unitaries with the Haar distribution. We observe that δ A (ρ U,α n+1 ) ≥ D A (ρ U,α n+1 ). Finally, let us determine the unitaries U n leading to the smallest discord. The fidelity (24) takes its maximal value 1 when e iω k ∈ {e iφ , −e iφ } ∀ k = 1, . . . , N , for a fixed angle φ ∈ [0, 2π[. Hence D A (ρ U,α n+1 ) = 0 if and only if U n = e iφ Q with φ ∈ [0, 2π[, Q = Q † , and Q 2 = Q. Fig. 2 displays the GQD for finite system sizes n. In agreement with the aforementioned criterium, the discord vanishes for rotation matrices U n with rotation angles equal to 0 or π. Moreover, it is independent of n when the rotation angle is equal π/2. The existence of output states of the DQC1 model with a zero discord and the necessary and sufficient condition stated above have already been discussed in Ref. [13]. Let us notice that the particular matrices U n which have been conjectured in [12] to lead for α = 1 to the highest possible bipartite entanglement among the (n + 1) qubits satisfy this condition. For such U n the discord between the control qubit and the n target qubits thus vanishes.
The eigenvalues of Λ(u) come in opposite pairs (λ ± (u), −λ ± (u)) with in agreement with tr Λ(u) = tr(ρ A σ u ) = 0. But |a m | ≤ b m by (36), thus the second square root is smaller than the first one. One deduces from (14) that If the b m are distinct from each other, the maximum is reached for cos 2 θ, cos 2 φ ∈ {0, 1}, that is, for u = ±e 1 , ±e 2 , or ±e 3 (here e m are the coordinate axis vectors).
We notice that if |c m | is maximum for m = m max then this is also true for b m , i.e. the maximal number inside the brackets in (40) is the (m max )th one. Actually, one can show from (36) and (30) that It is clear on (40) that the vectors c such that b max takes the highest possible value b max = 1 have a single non-vanishing component c m . Therefore, in agreement with the results of Ref. [13], the A-classical states with maximally mixed marginals are located (up to local unitary equivalence) on the three segments of the coordinate axes inside the tetrahedron, represented in Fig. 3(a) by thick dashed lines. As a consequence, all states ρ such that c is inside the octahedron formed by the convex hull of the three aforementioned segments are separable (for indeed, convex combinations of A-classical states are separable). It follows from the Peres-Horodecki criterium that this octahedron contains all separable states with maximally mixed marginals [35].
The states (29) with the highest discord D A = 1 have b 1 = b 2 = b 3 = 0. These states are the four Bell states |Ψ ν located at the vertices F ± and G ± of the tetrahedron (see Fig. 3(a)). Note that an analogous result holds for the Hilbert-Schmidt-GQD [13].
Let us end this subsection by remarking that for twoqubit states ρ with maximally mixed marginals ρ A = ρ B = 1/2, the inequality (10) is an equality. Actually, for such ρ one has equal prior probabilities η i = α i |ρ A |α i = 1/2 in the QSD task of Sec. II C. Moreover, it follows from (29) that ρ is invariant under conjugation by the spin-flip operators σ 1 ⊗ σ 1 and σ 2 ⊗ σ 2 . But it has been observed above that an optimal direction u opt is given by one of the coordinate vectors e m or its opposite. Disregarding irrelevant phase factors, the two eigenvectors |α opt 0 and |α opt 1 of σ u opt are transformed one into another by the spin-flip operator σ m , with m = 1 if u opt = ±e 2 , ±e 3 and/or m = 2 if u opt = ±e 1 , ±e 3 . Hence the states ρ i are related by a unitary conjugation, ρ 1 = σ m ⊗ σ m ρ 0 σ m ⊗ σ m . The square-root measurement is known to be optimal to discriminate such symmetric ensembles of states with equal prior probabilities [28,29]. Thus F A (ρ) is given by the right-hand side of Eq.(10). Based on this observation, one can rederive formula (39) in a slightly simpler way.
In Fig. 4, the four quantum correlation measures D A , δ A , D A , and E are plotted together for the Werner states, obtained by taking c 1 = c 2 = c 3 = −c in (29), with 0 ≤ c ≤ 1. Fig. 5(a) displays the difference D A (ρ) − δ A (ρ) for a subfamily of states with maximally mixed marginals depending on two parameters. We observe that In contrast, one sees in Figs. 4 and 5(b) that the GQD with Hilbert-Schmidt distance can be either smaller or larger than δ A , as it has been noted previously in other   Fig. 3(a)). From top to bottom: quantum discord δA (black boxes), GQD D

(2)
A with Hilbert-Schmidt distance (red crosses), GQD DA with Bures distance (blue circles), and geometric measure of entanglement E (green diamonds).
works [39]. We have looked numerically for vectors c inside the tetrahedron violating (43) and have not found any such vector. This indicates that this bound holds for any state ρ with maximally mixed marginals. It would be of interest to know if it also holds for more general states. Let us recall from Sec. IV that (43) is true for the output state of the DQC1 model with random unitaries.

C. Closest A-classical states
We now turn to the problem of finding the A-classical states σ ρ satisfying These closest A-classical states to ρ provide useful information on ρ and on the geometry of the set of quantum states. Such information, which is not contained in the discord D A (ρ), can be of interest when studying dynamical evolutions, for instance, when the two subsystems are coupled to their environments and undergo decoherence processes. Like in the two previous subsections, we focus on two-qubit states with maximally mixed marginals given by Eq. (29). The geometric representation of these states by vectors c in the tetrahedron T will give some A (ρ) between the quantum discord and the normalized Hilbert-Schmidt-GQD.
insight to our results. In order to compare the geometries on T associated to the two distances d B and d 2 , we also determine the closest A-classical states for the Hilbert-Schmidt distance d 2 by using the results of Ref. [13].
The states σ ρ are obtained by applying formula (11). In view of the results of Sec. III, the optimal basis i=0 is the eigenbasis of the spin operator σ u opt in the optimal direction u opt . We already know that for the states (29), u opt is parallel to the coordinate vector e mmax if the maximum |c| = max m=1,2,3 |c m | is reached for a single index m = m max (see Sec. V A); otherwise, u opt is an arbitrary unit vector in the space spanned by the e m corresponding to indices m such that |c m | = |c| (indeed, for such indices the parameters b m in (38) are equal, b m = b max , by virtue of (41)). Moreover, it has been shown in Sec. V A that the states ρ opt i are optimally discriminated by the square root measurement given by the projectors Let us first take u opt = ±e mmax . Then σ mmax = ±(|α opt 0 α opt 0 | − |α opt 1 α opt 1 |). One infers from the expression (32) of the matrix √ ρ in the standard basis that √ ρ = 3 ν=0 t ν σ ν ⊗ σ ν , where t ν are given by (33) and we have set σ 0 = 1. This yields with t max ≡ t mmax . Replacing (45) and (46) into (11) and using the identities F A (ρ) = 4(t 2 0 + t 2 max ) and 16t 0 t m = a m + c m , which follow respectively from (39) and (35) and from (33), (30), and (36), one finds where with analogous formulas for a 2 and a 3 obtained via a permutation of indices. Hence, a two-qubit state ρ with maximally mixed marginals always admits among its closest A-classical states a state with maximally mixed marginals, given by Eq. (47). This closest state has the following decomposition in terms of the four Bell states Interestingly, the closest states (49) for the Bures distance have the same form as the closest states for the relative entropy. Actually, the states σ ∆ ρ satisfying S(ρ||σ ∆ ρ ) = min σ∈CA S(ρ||σ) are given by Eq.(49) but with different probabilities q mmax and 1 − q mmax , given by the sums of the two largest or the two smallest probabilities p ν [37]. According to our notation, these two smallest (respectively largest) probabilities are p 0 and p mmax if |c m | = |c| for a single index m = m max and p 0 + p mmax < 1/2 (respectively p 0 + p mmax > 1/2).
If |c m | = |c| for two indices m, say, |c 1 | = |c 2 | > |c 3 | with c 1 = ±c 2 , all the eigenbasis {|α opt φ,i } of the spin operators σ φ = cos φ σ 1 + sin φ σ 2 maximize the success probability P opt v.N. (8). Hence ρ has infinitely many closest A-classical states σ ρ . These states can be determined by a simple generalization of the calculation leading to (47), choosing for Π opt i the squareroot measurement operators (45) associated to {|α opt φ,i }. Let us note that c 1 = ±c 2 implies t 1 = ±t 2 and recall the identities |α opt φ,i = e −i φ 2 σ3 |α opt 0,i and e i φ 2 σ3 σ u e −i φ 2 σ3 = σ u ′ , where u ′ is related to u by a rotation around the zaxis with the angle −φ. We find the following family of closest A-classical states depending on the optimal vector u = u 1 e 1 + u 2 e 2 with u 2 1 + u 2 2 = 1: with m max ∈ {1, 2} and u(c) = c −1 mmax (c 1 u 1 e 1 + c 2 u 2 e 2 ) (note that σ ρ (u) is independent of the choice of m max in {1, 2} since a 2 + c 2 = ±(a 1 + c 1 )). Similar results are obtained by permutation of indices in the cases |c 1 | = |c 3 | > |c 2 | and |c 2 | = |c 3 | > |c 1 |. The A-classical states (51) have maximally mixed marginals, although they are not of the form (29) save for u 1 = 0, ±1 (in fact, σ ρ (u) can be transformed into the state (47) by an appropriate conjugation by a local unitary operator). If |c 1 | = |c 2 | = |c 3 |, any orthonormal basis is optimal and one obtains a family of closest states depending on the arbitrary optimal unit vector u ∈ R 3 , given by (51) with u(c) = c −1 mmax (c 1 u 1 e 1 + c 2 u 2 e 2 + c 3 u 3 e 3 ). Let us now discuss a more subtle point, which is of relevance if one wants to find all A-classical states σ ρ satisfying (44). Given an optimal basis {|α opt i }, there is not guarantee that the square-root measurement (45) is the only optimal measurement maximizing P S ({ρ opt i , η opt i }). Actually, it turns out that this is not the case if c belongs to one of the faces of the tetrahedron (so that p 0 p 1 p 2 p 3 = 0). In this situation, we encounter in the appendix a whole family of optimal projectors Π opt i (r), depending on a real parameter r ∈ [−1, 1]. Let us emphasize that if two states of the form (11) with the same optimal basis {|α opt i } are closest A-classical states to ρ then, by convexity of the square Bures distance, any convex combination of these states is also a closest A-classical state to ρ. Hence the set of optimal projectors must be either infinite or reduced to one point. To each measurement in this set one can associate a closest A-classical state by Eq. (11), which is determined in the appendix. Before giving the result, let us introduce the product basis {|α i , β j = |α i ⊗ |β j } 1 i,j=0 of C 2 ⊗ C 2 defined as follows: (i) if |c m | is maximal for a single index m = m max , then |α i = |β i are the eigenvectors of σ mmax ; (ii) if |c m | is maximum for exactly two components c m , then m max , |α i , and |β i are defined by for some arbitrary angles φ or θ ∈ [0, 2π[; (iii) if |c 1 | = |c 2 | = |c 3 |, that is, c 1 = ǫ 2 c 2 = ǫ 3 c 3 with ǫ 2,3 ∈ {−1, 1}, then for arbitrary angles φ and θ ∈ [0, 2π[. Then all the closest A-classical states to ρ are of the following form: if p 0 p mmax = 0 and p m > 0 ∀ m = m max , and if p 0 p mmax > 0 and p 1 p 2 p 3 = 0. In these equations r ∈ [−1, 1] is arbitrary. When c is in the interior of the tetrahedron T (i.e., p 0 p 1 p 2 p 3 > 0), all the closest states are obtained by setting r = 0 in Eqs. (54) and (55). One then recovers the previous results (47) and (51). Therefore, ρ has a unique closest A-classical state if and only if c is in the interior of T and the maximum |c| is non-degenerate (case (i)). In the case (ii), ρ admits a 1-parameter (or, if c belongs to a face of T , a 2-parameter) family of closest A-classical states. In the case (iii), this family is a 2-parameter (or, if c belongs to a face of T , a 3-parameter) family. It should be noted that the states (54) and (55) do not have maximally mixed marginals save for r = 0. This means that the states ρ on the faces of T have closest A-classical states outside the tetrahedron. Let us also emphasize that all the closest A-classical states σ ρ determined in this section are in fact classical states, that is, the eigenvectors of σ ρ are product states. Thus as could be expected from the symmetry of ρ under the exchange of the two qubits. In Figs. 3(b) and 3(c), some examples of states ρ in the tetrahedron and their closest states σ ρ are represented. In Fig. 3(b), outside the dashed lines on which two or more components c m have equal moduli, σ ρ is specified by a vector s lying on the closest coordinate semi-axis to c for the usual distance in R 3 (but s is not the closest vector to c on that semi-axis).
The closest A-classical states for the Hilbert-Schmidt distance d 2 to a state ρ of the form (29) can be found by using the results of Ref. [13]. They are given by σ (2) ρ (u) = with an arbitrary unit vector u ∈ R 3 . In contrast with the Bures distance, all states ρ with maximally mixed marginals have infinitely many closest A-classical states. Moreover, there are three states σ (2) ρ of the form (29) lying on the coordinate axes, with non-vanishing coordinate equal to the corresponding coordinate of c, irrespective of the order of the moduli |c m |. Since tr[(σ are always mixed. In particular, unlike for the Bures distance, the closest classical states to the maximally entangled pure states |Ψ ν are not pure product states.

VI. CONCLUSIONS
We have shown that the results of Ref. [14] can be used to determine the geometric quantum discord (GQD) with Bures distance of a state ρ of a bipartite system AB and the corresponding closest A-classical states to ρ when the subsystem A is a qubit. Analytical expressions for this GQD in the DQC1 model and for two qubits in a state with maximally mixed marginals have been obtained. In all cases for which exact analytical formulas for the usual quantum discord of Refs. [4,5] are also available, we observe that the normalized GQD is smaller than the usual discord. In the DQC1 model, the comparison of the GQDs of the output state for different unitaries U n shows that the highest discord appears when U n has uniformly distributed eigenvalues on the unit disk modulo a symmetry with respect to the origin. In particular, the discord cannot be larger than that obtained for large random unitary matrices distributed according to the Haar measure, which have been studied in [7]. For rotation matrices and large number of qubits n, the GQD is close to this upper bound, expected for small rotation angles and at certain specific angles. For two-qubit states ρ with maximally mixed marginals, the closest A-classical states for the Bures distance are qualitatively similar to the closest A-classical states for the relative entropy, but completely different from those for the Hilbert-Schmidt distance. The geometry induced by the Bures distance in the tetrahedron looks like the Euclidean geometry of R 3 , the A-classical states being located on the three segments of the coordinate axes inside the tetrahedron. However, depending on the symmetry of ρ, ρ can have either a unique closest classical state with maximally mixed marginals or a continuous family of closest classical states with maximally mixed or nonmaximally mixed marginals.
Our results constitute a first step in the study of the geometric measure of quantum discord introduced in [14] in some concrete models and its relation with the usual discord and other geometrical versions. It would be of interest to calculate the Bures-GQD in other physical models, in particular in the presence of dynamical evolutions, and to compare it with other measures of non-classicality in the literature.
Note added: After the completion of this work we have been informed by G. Adesso that the Bures geometric discord for two-qubit states with maximally mixed marginals has been calculated independently in [40] by a different method, yielding to the same result (Eq. (39)).
Appendix A: Closest classical states of a two-qubit state with maximally mixed marginals In this appendix we determine all optimal projective measurements {Π opt i } maximizing the success probability P S ({ρ opt i , η opt i }) in Eq.(13) for a two-qubit state ρ of the form (29). This allows us to find all the closest A-classical states to ρ by applying formula (11).
Before starting the calculation, let us first note that we can restrict our analysis to optimal directions u opt satisfying u opt · e m ≥ 0 for a fixed coordinate index m = 1, 2, or 3. Indeed, changing the order of the vectors in the basis {|α opt i } 1 i=0 in Eqs. (9) and (11) amounts to exchange (ρ opt 0 , η opt 0 ) ↔ (ρ opt 1 , η opt 1 ) and Π opt 0 ↔ Π opt 1 . This does clearly not modify the optimal success probability and the A-classical states σ ρ . But {|α opt i } is the eigenbasis of the spin operator σ u for the optimizing direction u = u opt , see Sec. III. Changing the order of the basis vectors thus corresponds to invert u opt . Hence we may assume u opt · e m ≥ 0 without loss of generality. Thanks to the results of Sec. V, we know that if |c m | is maximal for a single index m = m max then u opt = e mmax .