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New Journal of Physics
The open access journal for physics
Geometric quantum discord with Bures distance
D Spehner1,2,4 and M Orszag3
1 Université Grenoble 1 and CNRS, Institut Fourier UMR5582, BP 74, F-38402 Saint Martin d'Heres, France
2 Universite Grenoble 1 and CNRS, Laboratoire de Physique et Modelisation des Milieux Condenses UMR5493, BP 166, F-38042 Grenoble, France
3 Pontificia Universidad Católica, Facultad de física, Casilla 306, Santiago 22, Chile
E-mail: Dominique.Spehner@ujf-grenoble.fr
New Journal of Physics 15 (2013) 103001 (18pp)
Received 10 April 2013 Published 2 October 2013 Online at http://www.njp.org/
doi:10.1088/1367-2630/15/10/103001
Abstract. 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 state p are obtained in terms of the corresponding optimal measurements.
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New Journal of Physics 15 (2013) 103001 1367-2630/13/103001 + 18$33.00
© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
Contents
1. Introduction 2
2. Definitions of the quantum discords and Bures distance 4
2.1. The quantum discord and the set of A-classical states............................4
2.2. Distance measures of quantum correlations with the Bures distance..............5
3. The Bures geometric quantum discord of pure states 7
4. The Bures geometric quantum discord of mixed states 9
4.1. Link with minimal error quantum state discrimination............................9
4.2. Derivation of the variational formula (23)........................................13
4.3. Closest A-classical states..........................................................14
5. Conclusions 15 Acknowledgments 16 Appendix. Necessary and sufficient condition for the optimal success probability to be
equal to the inverse number of states 16
References 17
1. 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 [5-8] or in some quantum communication protocols [8-11]. 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 dB on the convex cone of density matrices, which has better properties than the Hilbert-Schmidt distance d2 from a quantum information perspective. In particular, dB 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 d2. This entanglement measure has been widely studied in the literature [22-25]. By analogy with entanglement, a geometric measure of quantum discord has been defined by Dakic et al [14] as the minimal distance of the system state to the set of zero-discord states. This geometric quantum discord (GQD) has been evaluated explicitly for two qubits [14]. However, the aforementioned authors use the Hilbert-Schmidt distance d2, 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 dB, 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 p, and hence learn something about the 'position' of p 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 p is closely linked to a minimal error quantum state discrimination (QSD) problem.
The task of discriminating states pertaining to a known set (pi,..., pn} of density matrices pi with prior probabilities n plays an important role in quantum communication and quantum cryptography. For instance, the set (p1, ..., pn} can encode a message to be sent to a receiver. The sender chooses at random some states among the pt '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 pi are nonorthogonal, 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 paper [29]). In the first one, the receiver seeks for a generalized measurement with (n + 1) outcomes, allowing him to identify perfectly each state pi 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 PS = n= nP|i, where Pi\i is the probability of the measurement outcome i given that the state is pi . This strategy is called minimal error
(or ambiguous) QSD. The maximal success probability PSopt 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 pi are related to each other by some symmetry and have equal probabilities n (see [29, 32, 33] and references therein). However, several upper bounds on PSopt 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 p be any state of a bipartite system with a finite-dimensional Hilbert space. We will prove in what follows that the Bures-GQD of p is equal to the maximal success probability PSopt in the ambiguous QSD of a family of states {pi} and prior probabilities {n} depending on p. Moreover, the closest zero-discord states to p are given in terms of the corresponding optimal von Neumann measurement(s). The number of states pi 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 [ , ]: PSopt and the optimal von Neumann projectors are given in terms of the eigenvalues and eigenvectors of the Hermitian matrix A = n0p0 — nipi. 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 paper is organized as follows. The definitions of the quantum discords and of the Bures distance are given in section 2, together with their main properties. In section 3, 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 section 4. The last section contains some conclusive remarks and perspectives. The appendix contains a technical proof of an intuitively obvious fact in QSD.
2. Definitions of the quantum discords and Bures distance
2.1. The quantum discord and the set of A-classical states
In this work we consider a bipartite quantum system AB with Hilbert space H = HA ® HB, the spaces HA and HB of the subsystems A and B having arbitrary finite dimensions nA and nB. The states of AB are given by density matrices p on H (i.e. Hermitian positive N x N matrices p e Mat(C, N) with unit trace tr(p) = 1, with N = nAnB). The reduced states of A and B are defined by partial tracing p over the other subsystem. They are denoted by pA = trB (p) and p b = tr a (p).
Let us first recall the definition of the quantum discord [3, 4]. The total correlations of the bipartite system in the state p are described by the mutual information IA:B (p) = S(pA) + S(pB) — S(p), where S(•) stands for the von Neumann entropy. The amount JB|A(p) 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 {nA} of projectors acting on HA (i.e. by self-adjoint operators niA on HA satisfying niA= SijnA). Hence JB\A(p) = max^A}{S(pB) — J]i qiS(pB|i)}, where the maximum is over all von Neumann measurements {nA}, qi = tr(nA ® 1 p) is the probability of the measurement
outcome i, and pB= q-1 trA (niA ® 1 p) is the corresponding post-measurement conditional state of B. The quantum discord is by definition the difference 8A (p) = IA:B (p) — JB\A (p) 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 8 A (p) ^ 0 for any p. Moreover, 8 A (aA-cl) = 0 if and only if
aA-cl = qi la- ><a I ® ^b ii, (1)
where {|ai>}n= 1 is an orthonormal basis of HA, aB|i are some (arbitrary) states of B depending on the index i, and qi ^ 0 are some probabilities, i qi = 1. The fact that 8A (aA-cl) = 0 follows directly from IA:b (aA-cl) = s(trA (aA-cl)) — Y.iqis(ab|i) ^ Jb| a (aA-cl) and from the non-negativity of the quantum discord. For a bipartite system in the state aA-cl, the subsystem A is in one of the orthogonal states |ai > with probability qi, 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 CA the set of all A-classical states. By using the spectral decompositions of the aB^, any A-classical state aA-cl e CA can be decomposed as
aA-cl = J2 J2 qij |a ><a | ® |i XPj |i |, (2)
i = 1 j = 1
where, for any fixed i, j^>1^=1 is an orthonormal basis of HB, and qij ^ 0, J] j qij = 1 (note that the |i> need not be orthogonal for distinct i's). One defines similarly the set CB 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 {|ai> ® j>}n=fB=1 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 qij.
Let us point out that CA, CB and the set of classical states C are not convex. Their convex hull is the set S of separable states. A state asep is separable if it admits a convex decomposition asep = T^mqm ><0m | ® >№m |, where {|0m >} and {|^m>} are (not necessarily orthogonal) families of unit vectors in HA and HB and qm ^ 0, m qm = 1. For pure states, A-classical and B-classical, classical and separable states all coincide. Actually, according to (2) the pure A-classical (and, similarly, the pure B-classical) states are product states.
2.2. Distance measures of quantum correlations with the Bures distance
The GQD of a state p of AB has been defined in [14] as the square distance of p to the set CA of A-classical states
Df (p) = d2(p, Ca)2 = min d2(p, aA-cl)2, (3)
aA-cl eC a
where d2(p,a) = (tr[(p — a)2])1/2 is the Hilbert-Schmidt distance. Instead of taking this distance, we use in this paper the Bures distance
dB(p, a) = [2(1 — VF(p, a))j5, (4)
where p and a are two density matrices and F(p, a) is their fidelity [1, 18, 41]
f (p, a) = ii vpva 112 = [tr([vap va]1/2 )]2. (5)
It is known that (4) defines a Riemannian distance on the convex cone E C 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, dB satisfies the following properties [1, 41]: for any p, a, p1, p2, a1 and a2 e E,
(i) joint convexity: if n1, n2 ^ 0 and n1 + n2 = 1, then dB (n1 p1 + n2p2, n1 a1 + n2a2) ^ m dB (p1, a1) + n2d (p2, a2) and
(ii) dB is monotonous under the action of completely positive trace-preserving maps T from Mat(C, N) into itself: for any such T, dB(Tp, Ta) ^ dB(p, a).
Property (ii) implies that dB is invariant under unitary conjugations: if U is a unitary operator on H, then dB(UpUf, UaUf) = dB(p, a). Note that the Hilbert-Schmidt distance d2 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 d1 (p, a) = tr(|p — a |) as follows [1]:
dB(p, a)2 ^ d1 (p, a) ^ [1 — (1 — 1 dB(p, a)2)2]2 . (6)
For good reviews on the Uhlmann fidelity and Bures distance, see the book of Nielsen and Chuang [1] and the nice introduction of the paper [43] devoted to the estimation of the Bures volume of E.
We define the GQD as
Da (p) = dB (p, Ca )2 = 2(1 —J Fa (p)), Fa (p) = max F (p,aA-cl). (7)
aA-cleC A
The unitary invariance of dB and d2 implies that DA and D^ are invariant under conjugations by local unitaries, p ^ UA ® UBp UA ® UB, since such transformations leave CA invariant. By property (ii), DA 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
D(p) = dB(p, C)2 = 2(1 — yFC(p)), E(p) = dB(p, S)2 = 2(1 — TFS(p)), (8)
where FC (p) is the maximal fidelity between p and a classical state acl e C and FS(p) the maximal fidelity between p and a separable state asep e 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]
Fs(p) = max VpmFs(|^m», Fs(|^m» = max F(|^m|,asep), (9)
{pm},{|^m>} ^ asepeS
where the maximum is over all pure state decompositions p = m Pm |*m}(*m I of p (with ||*m || = 1 and pm ^ 0, Pm = 1). 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
EEoF (p) = min V" Pm EEoF (| *m } ),
{Pm },{|*m
E EoF (|*m }) = S (tr a (|*m }(*m I)) = S (tr B (|*m }(*m I)).
Since C c CA C S, the three distances are ordered as E (p) ^ Da (p) ^ D (p).
This ordering of quantum correlations is a nice feature of the geometric measures. In contrast, the entanglement of formation EEoF (p) can be larger or smaller than the quantum discord Sa(p) [46, 47].
3. 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 DA in terms of the Schmidt coefficients pi. We recall that any pure state |*} e HA ® HB admits a Schmidt decomposition
|*} = E VP to}, (12)
where n = min{nA, nB} and {|^i}}n= 1 (respectively {|xj}}n=1) is an orthonormal basis of HA (HB). If the pi are non-degenerate, the decomposition (12) is unique, the pi and } (respectively |xj}) being the eigenvalues and eigenvectors of the reduced state (|*}(*|)A (respectively (|*}(*|)B). Note that p ^ 0 and J2i Pi = II*II2 = 1.
Theorem 1. If p* = |*}(* | is a pure state, then
Da(p*) = D(p*) = E(p*) = 2(1 - VPmax), (13)
where pmax is the largest Schmidt eigenvalue pi. If this maximal eigenvalue is non-degenerate, the closest A-classical (respectively classical, separable) state to p* is the pure product state v = |^max ® Xmax}(^max ® Xmax|, where |^max} and |xmax} are the eigenvectors corresponding to pmax in the decomposition (12). If pmax is r-fold degenerate, say pmax = p = • • • = pr > pr+i, ..., pn, then infinitely many A-classical (respectively classical, separable) states v minimize the distance dB(p*,v). These closest states v are convex combinations of the orthogonal pure product states |a/ ® fa}(al ® fa l l = 1, ..., r, with |a} = ^ri=1 uil } and } = r= 1 u*l|xi}, where (uil)ril=1 is an arbitrary r x r unitary matrix and } and |xi} are some eigenvectors in the decomposition (12).
The expression (13) of the geometric measure of entanglement E(p*) 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), DA(p*) = 0 if and only if |*} 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 inequality pmax ^ 1/n
(following from J2^=1 X = 1) one deduces that DA(p^) ^ 2(1 — 1/^/ft). The maximal value of Da is reached when xi = 1/n for any i, that is, for the maximally entangled states (recall that such states are the pure states with reduced states (p^)A and (p^)B having a maximal entropy S((p^)A) = — Yin=1 ¡¡i ln ¡i = ln(n)). Note that when ¡max is r-fold degenerate, the r vectors |a > (respectively >) are orthonormal eigenvectors of [p^ ] A (respectively [p^ ] B ) with eigenvalue ¡max. One then obtains another Schmidt decomposition of > by replacing in (12) the r eigenvectors > and |x > with eigenvalue ¡max by |a > and >.
Remarkably, the maximally entangled states are the pure states admitting the largest family of closest separable states (this family is a (n2 + n — 2) real-parameter submanifold of E). For instance, in the case of two qubits (i.e. for nA = nB = n = 2), the Bell states |0±> = (|00> ± |11»A/2 admit as closest separable states the classical states
a±= J2 qila)(«/|®|ßi)(ß,I, la)= u0/|0> + uu11>, |ß,) = uOl|0>± u 1
U 0lU 0m + u\iu 1m
= 8mi and qi ^ 0, qo + qi = 1. Interestingly, typical decoherence processes such as pure phase dephasing transform p0± into one of its closest separable state (|00>(00| + 111 >< 111)/2 at times t ^ tdec, where tdec is the decoherence time. Slower relaxation processes modifying the populations in the states |00> and 111 > do not further increase the distance to the initial state p$±. The situation is different for a partially entangled state > = 100> + 111 > with / > /0: then the closest separable state is the pure state 111 >, but > evolves asymptotically to a statistical mixture of |00> and 111 > when the qubits are coupled e.g. to thermal baths at positive temperatures.
Proof. For a pure state p^, the fidelity reads F(p^, aA_cl) = (^|aA_cl>. Replacing aA_cl in (7) by the right-hand side of (2) we get
Fa (p*) = max
{la )},{|ß j li )},{qij}
Y qij |(a ® ßj\i I*)|:
= max |(a ® ß|*)|2,
M=llßll=1
where we have used J]qtj = 1. Thanks to the Cauchy-Schwarz inequality, for any normalized vectors |a> g HA and > g Hb one has
|(a ® ß|*)| =
Y V^i (a to )(ß |Xi )
^ Y VXi |(a to )(ß |Xi )| ^ V^max ^|(a to )(ß |x )| i1
Y |(ato
Y |(ß |Xi
Let us first assume that / = /max > /2, ..., /n. Then |<a ® ^>| = ^/max if and only if |a> = to > and > = |x1 > up to irrelevant phase factors. Thus the maximal fidelity FA(p^) between p^ and an A_classical state is simply given by the largest Schmidt eigenvalue /max. Moreover, the maximum in the second member of equation (15) is reached when a single q^ is non_vanishing, say q^ = St 18j 1, and |a1 > = to >, |^1|1 > = |x1 >. This means that the closest A_ classical state to p^ is the pure product state to ® x ><^1 ® X1. Since this is a classical state,
one has FC(p*) = FA(p*) = pmax. One shows similarly that FS(p*) = pmax. Then (13) follows from the definitions (7) and (8) of DA, D and E.
More generally, let p1 = • • • = pr = pmax > pr+1,..., pn. We need to show that all inequalities in (16) and (17) are equalities for appropriately chosen normalized vectors |a} and }. The first inequality in (16) is an equality if and only if arg((a|xi}) = 0 is independent of i. The second inequality in (16) is an equality if and only if |a} belongs to ymax = span{|^i }}i=1 or } belongs to Wmax = span{|xi }}i=1. The Cauchy-Schwarz inequality in (17) is an equality if and only if |(a }| = |xi }| for all i, with A ^ 0. 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. |a} e span{|^i}}ni=1 and } e span{|xi}}ni=1 (this holds trivially if nA = nB = n). Putting all conditions together, we obtain |a} e Vmax, } e Wmax and (fa|xi} = e?0(^|a} for i = 1,..., r. Therefore, from any orthonormal family {|al }}r=1 of Vmax one can construct r orthogonal vectors |al ® fal} satisfying |(al ® fal |*}| = ^pmax for all l = 1, ..., r, with } = |al}. The probabilities {qij} maximizing the sum inside the brackets in (15) are given by qij = qi if i = j ^ r and zero otherwise, where {ql }r=1 is an arbitrary set of probabilities. The corresponding A-classical states with maximal fidelities F(p*,v) are the classical states v = J]r= 1 ql ® }(al ® |. □
The equality between the correlation measures DA, D and E is a consequence of the fact that the closest states to p* are classical states. Such an equality is reminiscent from the equality between the entanglement of formation EEoF and the quantum discord 8A 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
d2(p*, VA-cl)2 = tr[(|*}(*| - VA-cl)2] = 1 - 2F(p*, VA-cl) + tr(aA-d) (18) that the closest A-classical state results from a competition between the maximization of the fidelity F (p*, vA-cl) and the minimization of the trace tr(vA-cl), which is maximum for pure states. For instance, one can show [39] that the closest A-classical states to the Bell states |0±} for d2 are mixed two-qubit states. The validity of theorem 1 is one of the major advantages of the Bures-GQD over the Hilbert-Schmidt-GQD.
4. The Bures geometric quantum discord of mixed states
4.1. Link with minimal error quantum state discrimination
The determination of DA (p) 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 pi drawn from a known family {pi}n= 1 with prior probabilities {ni}n= 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 p j when his measurement result is j. The generalized measurement is given by a family of positive operators Mi ^ 0 satisfying i Mi = 1 (POVM). The probability to find the result j is Pj ^ = tr(Mj pi) if the system is in the state pi. The maximal success probability of the receiver reads
PSopt({pi}) = max T ntr(Mp). (19)
S POVM {Mi}
Theorem 2. Let p be a state of the bipartite system AB with Hilbert space H = H A ® HB and let a = {\ai)}n= 1 be a fixed orthonormal basis of HA. Consider the subset CA(a) C CA of all A-classical states aA-cl such that a is an eigenbasis of trB(aA-cl) (i.e. CA(a) is the set of all states aA-cl of the form (1), for arbitrary probabilities qi and states aB|i on HB). Then the maximal fidelity F (p, CA (a)) = maxaA-cl gCa (a) F (p, aA-cl) of p to this subset is equal to
F(p, Ca(a)) = Ps°PtVN ({pi, n}) = max £ ntr(npi), (20)
where PS°ptv N'({pi, n}) is the maximal success probability over all von Neumann measurements given by orthogonal projectors ni of rank nB (that is, self-adjoint operators on H satisfying ni n j = Sij ni and dim(niH) = nB), and
m = (a/|pa|ai), pi = n-1 Vp|ai)(ai| ® 1Vp (21)
(if ni = 0 then pi is not defined but does not contribute to the sum in (20)).
This theorem will be proven in section 4.2. Note that the pi are quantum states of AB if n > 0, because the right-hand side of the last identity in (21) is a non-negative operator and n is chosen such that tr(pi) = 1. Moreover, {n}n= 1 is a set of probabilities (since n ^ 0 and n = tr(p) = 1) and {pi, ni }n== 1 defines a convex decomposition of p, i.e. p = i n pi. Let us assume that p is invertible. Then the application of a result by Eldar [48] shows that the POVM maximizing the success probability PS ({pi, n}) in (19) is a von Neumann measurement with projectors ni of rank nB, i.e.
F (p, Ca (a)) = PsoptvN({pi, ni}) = Ps^ ({Pi, ni}), p > 0. (22)
In fact, one may first notice that all matrices pi have rank ri = nB (for indeed, pi has the same rank as nip-1/2pi = |ai)(ai | ® 1 Vp and the latter matrix has rank nB). Next, we argue that the pi are linearly independent, in the sense that their eigenvectors |§ij) form a linearly independent family {|§ijof vectors in H. Actually, a necessary and sufficient condition for |§ij-) to be an eigenvector of pt with eigenvalue kj > 0 is |£7) = (Aijni)-1 Vp|ai) ® IZij), IZij) ^ Hb being an eigenvector of Ri = (ai |p |ai) with eigenvalue kij ni > 0. For any i, the Hermitian invertible matrix Ri admits an orthonormal eigenbasis {|Zij)}nj=1. Thanks to the invertibility of
Vp, {\&j)}ijir11''nnB is a basis of H and thus the states pi 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 p of the bipartite system AB, the fidelity to the closest A-classical state is given by
Fa(p) = max max tr[n Vp\ai)(ai \ ® 1 Vp], (23)
{\ai)} {ni} ^ i=1
where the maxima are over all orthonormal basis {\ai)} of HA and all orthogonal families {ni }?= 1 of projectors of H A ® H B with ranknB. Hence, using the notation of theorem 2,
Fa (p) = max Ps^'dp,-, n}). (24)
{\ai)}
If p > 0 then one can replace psoptv N- [n (24) by the maximal success probability (19) over all POVMs.
It is noteworthy to observe that the basis vectors |ai) can be recovered from the states pi and probabilities n by forming the square-root measurement operators Mi = np-1/2pip-1/2, with P = Ei nPi (we assume here p > 0). Actually, such measurement operators are equal to the rank-nB projectors Mi = |ai )(a | ® 1. By bounding from below Psopt v • ({pi, ^}) by the success probability corresponding to ni = Mi, we obtain
Fa(p) ^ maxVtrB [(a |a)2]. (25)
{|ai)} T"" i=1
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 p.
(i) If p is an A-classical state, i.e. if it admits the decomposition (1), then the basis {|ai)} maximizing the optimal success probability in (24) coincides with the basis appearing in this decomposition. With this choice, one obtains n = qi and pi = |ai)(ai | ® aB|i for all i such that qi > 0. The states pi are orthogonal and can thus be perfectly discriminated by von Neumann measurements, so that FA (p) = PSoptv N ({pi, n}) = 1. Reciprocally, if FA (p) =
1 then PSoptv-R({A , nt}) = 1 for some basis {|a )} of H a and the corresponding p must be orthogonal, that is, pt = ntpt nt for some orthogonal family {nt} of projectors with rank nB. Hence p = E nt Pt = Et nt nt pt nt, Vp = Et nt Vp nt and (21) entails nt Pt = nt nt pt nt = Vp la )(a I ® iVp = Vpn |a )(a | ® 1 nt Vp, implying nt = |a )(a | ® 1 if p is invertible. Thus p is A-classical (this was of course to be expected since Da (p) = 0 if and only if p is A-classical, see section 2). Therefore, we can interpret our result (24) as follows: the non-zero-dtscord states p are such that the states (21) are non-orthogonal and thus cannot be perfectly dtscrtmtnated for any orthonormal basts {|at )} of H A.
(ii) If p = p* is a pure state, then all pt with ni > 0 are identical and equal to p^, so that Psopt = Psoptv-N- = sup{nt} Et nt (*n ) = nmax. One gets back the result Fa(p*) = ^max of section 3 by optimization over the basis {|at)}.
(iii) Let us determine the states p having the highest possible GQD, i.e. the smallest possible fidelity Fa(p).
Proposition. IfnA ^ nB, the smallest fideltty FA(p) for all states p of AB ts equal to 1/nA. If rnA ^ nB < (r + 1)nA wtth r = 1, 2, ..the states p wtth FA(p) = 1/nA are any convex combtnattons of the r maxtmally entangled pure states ) = n-1/2 EnAi ) ® ), k = 1, ..., r, wtth |0f ) = 5t7 and (f(k)) = Skl8tj.
We deduce from this result that the GQD DA (p) varies between 0 and 2 — 2/ VnA when nA ^ nB. By virtue of theorem 1, the proposition, and the inequality E(p) ^ DA(p), the geometric measure of entanglement E(p) also varies between these two values. This means that the most dtstant states from the set of A-classtcal states CA are also the most dtstantfrom the set of separable states S .If nA ^ nB < 2nA, these most distant states are always maximally entangled pure states.
Proof. The success probability psoptv N- must be clearly larger than the highest prior probability nmax = maxt{nt}. (A receiver would obtain Ps = nmax by simply guessing that his state is pt ,
with nmax = Птах; a better strategy is of course to perform the von Neumann measurement (П} such that nimax projects on a nB-dimensional subspace containing the range of pimax; this range has a dimension ^ nB by a similar argument as in the discussion following theorem 2.) In view of (24) and by using nmax ^ 1/па (since ni = 1) we get
fa (p) ^ — (26)
for any mixed state p.
When nA ^ nB the bound (26) is optimum, the value 1 /nA being reached for the maximally entangled pure states, see section 3. Thus 1/nA is the smallest possible fidelity. Let p be a state having such a fidelity FA (p) = 1/nA. According to (24) and since it has been argued before that PSoptv N ' ^ nmax ^ 1/nA, Fa(p) = 1/nA implies that PSoptv N '((pi, ni}) = 1/nA whatever the orthonormal basis (|ai>}. It is intuitively clear that this can only happen if the receiver gets a collection of identical states pi with equal prior probabilities n = 1/nA. A rigorous proof of this fact is given in the appendix. From (21) and p = J] npi we then obtain (a |pA |ai> = 1 /nA and pi = p for any i = 1, ..., nA and any orthonormal basis (|ai>}. The first equality implies pA = 1/nA. By replacing the spectral decomposition p = pk|^k>(^k| into (21), the second equality yields trB(|^k>(Ф/1) = n—18kl for all k, l with pkpi = 0. Taking advantage of this identity for k = l, one finds that the eigenvectors |^k> of p with positive eigenvalues pk have all their Schmidt eigenvalues equal to 1/nA, that is, their Schmidt decompositions read |^k> = n—1/2 ЕП= i 1ФР> ® l^/k)>. Moreover, trB№I) = 0 is equivalent to vBk)±VB° with vBk) = span(|^(k) >}ПА 1 С H b .If nB < (r + 1)па then at most r subspaces VB) may be pairwise orthogonal. Thus at most r eigenvalues pk are non-zero. □
Let us now discuss the case nA > nB .In that case the smallest value of the maximal fidelity FS(p) to a separable state is equal to 1/nB and FS(p) = 1/nB when p is a pure maximally entangled state. This is a consequence of (9) and of the bound FS (p^) ^ 1/nB for pure states p^ (see section 3). As a result, the geometric measure of entanglement E(p) varies between 0 and 2 — 2/^Л with n = min(nA, nB}, in both cases nA ^ nB and nB > nA. We could not establish a similar result for the GQD DA (p). When nA > nB, the bound (26) is still correct but it is not optimal, i.e. there are no states p with fidelities FA (p) equal to 1/nA. Indeed, following the same lines as in the proof above, one shows that if FA (p) = 1 /nA then the eigenvectors |^k> of p with non-zero eigenvalues must have maximally mixed marginals [p^k]A = 1 /nA. But this is impossible since rank([p^k]A) ^ nB by (12). According to the results of section 3, pure states p^ have fidelities FA (p^) ^ 1/nB, so one may expect that states close enough to pure states have fidelities close to 1/nB or larger. This can be shown rigorously by invoking the bound
F " + 1 -||PII nB — ^ (27)
Fa (p) ^-+--, (27)
nB nA nB
where ||p|| is the norm of p and ¿p = 0 if rank(p) ^ nB and 1 otherwise. This bound can be established as follows. Let us write p = p| + (1 — p)p' where ) is the eigenvector of p with maximal eigenvalue p =||p|| and the density matrix p' has support on [C|^. Choosing an orthonormal family (ni} of projectors ofrank nB satisfying n1) = ), we get from (24)
Fa (p) ^ Y ni tr(ni pi ) = p (ai |tr b )(* |)|ai ) + (1 — p ) Y ^ tr(n pi ) (28)
with n p7 — Vp7\®i )(a \® ^VP7 and n = (a\pA\a ). Let us fix the orthonormal basis [\ai)} such that la) is the eigenvector with maximal eigenvalue ^max in the Schmidt decomposition (12) of ). This leads to the maximal possible value p^max of the first term in the right-hand side of (28). We now bound the sum in this right-hand side by its imth term nmaxtr(nimp7m), where im is the index i such that n7 is maximum, i.e. n7 — nmax. If im > 1, one can find orthogonal projectors n1 and nim such that ) e n1 H and p7mH C nimH C [C\^ (recall that the p7 have ranks ^ nB). If im — 1, we choose n1 — \ + ni where ni is the
spectral projector of pi associated to the (nB — 1) highest eigenvalues q1 ^ q2 ^ • • • ^ 4nB-1. In all cases, tr(nimp7m) ^ 1 — q'nB. If rank(p) ^ nB then q'nB — 0, otherwise we bound q'nB by 1/nB (since £n—1 q7j — 1). Collecting the above results and using the inequalities ^max ^ 1/nB and nmax ^ 1/nA (since £nt 1 n7 — 1), one gets (27). Note that this bound is stronger than (26) only for states p satisfying ||p(1 + nA — nB)—1 or rank(p) ^ nB. In summary, we can only conclude from the analysis above that when nA > nB the smallest possible fidelity minpeE FA (p) lies in the interval (1/nA, 1/nB].
4.2. 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 — maxU \tr(UT) \, the maximum being over all unitary operators on H. Using also (2), one gets
yjF(p, aA-cl) — max\tr(UVpVaA-cl)\
— max| V ^qTjtr(U Vp \a)(a \ ® \faj\i)(faj\i \)
— maxE Vqj (°ij\Vp\a ® faj\i)
— max V \(^ij \Vp\a ® faj\i)\. (29)
In the third line we have replaced the maximum over unitaries U by a maximum over all orthonormal basis [\Oij)} of H (with \Oij) — Uf \ai ® faj\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 \O ij) in such a way that (Oij\Vp\ai ® faj\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 [\faj\i)} of HB and all set of probabilities [qij}. The maximum over the probabilities qij is easy to evaluate by using the Cauchy-Schwarz inequality and Ei j qij — 1. It is reached for
\(Oij \Vp\a ® fa j\i )\
Jlij \(Oij \Vp\a ® fa j\i )\
We thus obtain
«a = ^ rzzTa, ,2. (30)
F (p, CA (a)) = max max F (p,aA_d) = max max V" |(^j ^ |ßj ^ )|2, (31)
{|ßjii)} {qij} {|ßjii)}{|$ij)} ~'
where we have set ^ > = (at iVP iO^- > e H B. We proceed to evaluate the maximum over {Ißjit>} and (Oj>}. Let us fix i and consider the orthogonal family of projectors of H of rank n B defined by
nt = £ |Oij>(Oij|. (32)
By the Cauchy-Schwarz inequality, for any fixed i one has
max VW;it ißjit>|2 ^ Y IIit "2 = tr[n VPlat>(at l ® 1 VP]- (33)
(iß j it ^ jj
Note that (33) is an inequality if the vectors i^j t > are orthogonal for different j 's. We now show that this is the case provided that the iO tj> are chosen appropriately. In fact, let us take an arbitrary orthonormal basis (i Otj >} of H and consider the Hermitian nB x nB matrix S(i) with coefficients given by the scalar products S^ = (^j^^>. One can find a unitary matrix V(i) such that S(i) = (V(i)}tS(i) V(i) is diagonal and has non-zero diagonal elements in the first rt raws, where rt is the rank of S(t). Let O > = £nB 1 V() iOtl>. Then (i >} is an
___r^s r^s r^s r^s
orthonormal basis of H and J]j i°ij>(°ij i = nt. Moreover, the vectors iOjit> = (at i VPi°ij> = Yin= 1 Vfj) it> form an orthogonal set (iOjit>Yj=1 and vanish for j > rt. Therefore, for any fixed orthogonal family (ni }n= 1 of projectors of rank nB, there exists an orthonormal basis (i Otj >} of H such that (32) holds and the inequality in (33) is an equality. Substituting this equality into (31), one finds
F(p, Ca(a)) = max £ ||°j^ ||2 = max £ tr[n VPi«/>(af i ® 1 VP], (34)
(ni } (ni }
i, J i
which yields the result (20). The formula (23) is obtained by maximization over the basis (iai>}. □
4.3. 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 P. To this end, one must find the orthonormal basis (iaiopt>} of HA maximizing PSoptvN'((Pi, n}) in (24) and the optimal von Neumann measurement (nopt} yielding the minimal error in the discrimination of the ensemble (Piopt, n°°pt} associated to (iaiopt>} in equation (21).
Theorem 4. The closest A-classical states to P are
= E |a0pt><a0ptI ® (a,0ptivpn0ptVPla0pt>, (35)
Fa (p) f=1
where {|aopt>}•== 1 and {nOpt}"== i are such that Fa(p) = tr^VP|a°pt>(aOpt| ® 1 vp] (see equation (23)).
Proof. This follows directly from the proof of the previous subsection. Actually, by using the expression (30) of the optimal probabilities qij and the fact that the Cauchy-Schwarz
inequality (33) is an equality if and only if ^) = ^)/||^j^1| when||^j|| = 0, we conclude that the closest A-classical states to p are given by (1) with |a) = |a°pt) and
^ , ZTj=1 j")(7ji1 (36)
qiOb|i = 2^qijlPjli)(Pjli1 = V-"A II7 112• (36)
j=1 =1 j=1 Wrj |i II
The denominator is equal to FA(p), see (34). The numerator is the same as the second
^ oot ^
factor in the right-hand side of (35). For indeed, by construction 17j|i) = (ai l\/~Pl7ij) and
Ej l7ij)(7ijl = nopt.
Let us stress that the optimal measurement {nopt} and basis {|a°pt)} may not be unique, so that p may have several closest A-classical states op. This is the case for instance when p is a pure state with a degenerate maximal Schmidt eigenvalue, as we have seen in theorem 1. If op = E qt la°pt)(a°pt | ® oB|i and op = e qi-|aiopt)(a°pt l ® oare two closest A-classical states to p with the same basis {|a°pt)} then so are all convex combinations op(n) = nop + (1 — n)op with 0 ^ n ^ 1. This fact is a direct consequence of the convexity of the Bures distance (property (i) in section 2), given that op(n) e CA. As a result, states p having more than one closest A-classical state will generally admit a continuous family of such states.
5. Conclusions
We have established in this paper a link between ambiguous QSD and the problem of finding the minimal Bures distance of a state p of a bipartite system AB to a state with vanishing quantum discord. More precisely, the maximal fidelity between p and an A-classical (i.e. zero discord) state coincides with the maximal success probability in discriminating the nA states popt with prior probabilities n°°pt given by equation (21), nA being the space dimension of subsystem A (theorem 3). These states and probabilities depend upon an optimal orthonormal basis {|a°pt)} of A. The closest A-classical states to p are, in turn, given in terms of this optimal basis and of the optimal von Neumann measurements in the discrimination of {popt, n°°pt} (theorem 4). Finally, we have shown that when nA ^ nB, the 'most quantum' states characterized by the highest possible distance to the set of A-classical states are the maximally entangled pure states, or convex combinations of such states with reduced B-states having supports on orthogonal subspaces. These states are also the most distant from the set of separable states.
As stated in the introduction, the QSD task can be solved for nA = 2 states. Thus the aforementioned results provide a method to find the geometric discord DA and the closest A-classical states for bipartite systems composed of a qubit A and a subsystem B with arbitrary space dimension nB ^ 2. In particular, explicit formulae can be derived for two qubits in states with maximally mixed marginals and for (nB + 1)-qubits in the DQC1 algorithm [39]. For subsystems A with higher space dimensions nA > 2, several open issues deserve further studies. Firstly, it would be desirable to characterize the 'most quantum' states when nA > nB. 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 toward classical states. In fact, our analysis may allow in some cases to determine the geodesic segment linking a given state p0 with a non-zero discord to its closest A-classical state aP0. Such a piece of geodesic is contained in the set of all states p having the same closest A-classical state ap = aP0 as p0. It would be of interest to compare in specific physical examples these Bures geodesics with the actual paths followed by the density matrix during the dynamical evolution.
Acknowledgments
We are grateful to Maria de los Angeles Gallego for revising the derivation of equation (23) and for providing us some notes about the calculation of the geometric discord for Werner states. DS 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 Analisis estocastico y aplicaciones'.
Appendix. Necessary and sufficient condition for the optimal success probability to be equal to the inverse number of states
Let {pi }n= 1 be a family of nA states on H with prior probabilities n, where nA = N/nB is a divisor of dim(H) = N. We assume that the pi have ranks rank(pi) ^ nB for any i. Let PSoptv N ({pi, ni}) be the optimal success probability in discriminating the states pi, defined by equation (20). We prove in this appendix that PSoptv N ({pi, n}) = 1 /nA if and only if n = 1 /nA for any i and all states pi are identical.
The conditions n = 1/nA and pi = p are clearly sufficient to have PSoptv N ({pi, n}) = 1/nA (a measurement cannot distinguish the identical states pi 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 PSoptvN ({pi, n}) = 1/nA. The equality n = 1/nA for all i is obvious from the bounds PSoptvN ({pi, n}) ^ nmax = maxi{n} and nmax ^ 1/nA (see section 4.1). Therefore, according to our hypothesis, any orthogonal family {ni 1 of projectors of rank nB satisfies J]i tr(nipi) = nAPS({pi, n}) ^ 1. We now argue that the states pi have ranges contained in a common subspace V. In fact, let V be the nB-dimensional subspace of H spanned by the eigenvectors of p1 associated to the nB highest eigenvalues (including degeneracies), and let us denote by n1 the projector onto V. Then p1 H C V (since we have assumed rank(p1) ^ nB) and thus p1 = n1 p1. Thanks to the inequality above, 1 i tr(nipi) ^ tr(n1 p1) = 1. It follows that tr(n2p2) = 0 for any projector n2 of rank nB orthogonal to n1. Hence p2, and similarly all pi, i = 3,... , nA, have ranges contained in V. This proves the aforementioned claim.
In order to show that all the states pi are identical, we further introduce, for each 1 ^ k ^ nB, some nB-dimensional subspace V(k) containing the eigenvectors associated to the k highest eigenvalues A1 ^ • • • ^ Ak of p1, the other eigenvectors being orthogonal to V(k) (then V(nB) = V). We also choose a nB-dimensional subspace W(k) C H orthogonal to V(k) such that W(k) 0 V(k) D V. Let {n(k)}n= 1 be an orthogonal family of projectors of rank nB such that nf) and n2k) are the projectors onto V(k) and W(k), respectively. Then
1 ^ E tr(n(k)Pi) = tr(n1k)p1) +tr[(1 - nf))p2] = 1 + A 1 + ... + Ak - tr(nf)p2), (A. 1 )
where we used £f n(*> = 1 and pH C W(k) 0 V(k) in the first equality. By virtue of the minmax theorem, tr(nf )p2) is smaller than the sum of the k highest eigenvalues of p2 (including degeneracies). By (A.1), this sum is larger than the sum A + • • • + Ak of the k highest eigenvalues of p1. By exchanging the roles of p1 and p2, we obtain the reverse equality. Since moreover k is arbitrary between 1 and nB, it follows that p1 and p2 have identical eigenvalues. By using (A.1) again, tr(n1k)p2) is equal to the sum of the k highest eigenvalues of p2. Hence the k corresponding eigenvectors of p2 are contained in the k-dimensional subspace V(k) 0 V. Since k is arbitrary, this proves that p1 and p2 have identical eigenspaces. Therefore p1 = p2. Repeating the same argument for the other states pi, i ^ 3, we obtain p1 = • • • = pnA.
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