Quantum discord for two-qubit X-states : A comprehensive approach inspired by classical polarization optics

# Quantum discord for two-qubit X-states : A comprehensive approach inspired by classical polarization optics

Krishna Kumar Sabapathy Optics & Quantum Information Group, The Institute of Mathematical Sciences, C.I.T Campus, Tharamani, Chennai 600 113, India.    R. Simon Optics & Quantum Information Group, The Institute of Mathematical Sciences, C.I.T Campus, Tharamani, Chennai 600 113, India.
###### Abstract

Classical correlation and quantum discord are computed for two-qubit -states. Our approach, which is inspired by the methods of classical polarization optics, is geometric in the sense that the entire analysis is tied to the correlation ellipsoid of all normalized conditional states of the A-qubit with measurement elements applied to the B-qubit. Aspects of the computation which depend on the location of the reduced state of A inside the ellipsoid get clearly separated from those which do not. Our treatment is comprehensive : all known results are reproduced, often more economically, and several new insights and results emerge. Detailed reexamination of the famous work of Ali, Rau, and Alber [Phys. Rev. A 81, 042105 (2010)], in the light of ours and a counterexample we manufacture against their principal theorem points to an uncommon situation in respect of their principal result : their theorem turns out to be ‘numerically correct’ in all but a very tiny region in the space of -states, notwithstanding the fact that their proof of the theorem seems to make, in the disguise of an unusual group theoretic argument, an a priori assumption equivalent to the theorem itself.

###### pacs:
03.65.Ta,03.67.-a,42.25.Ja

## I Introduction

The study of correlations in bipartite systems has been invigorated over the last couple of decades or so. Various measures and approaches to segregate the classical and quantum contents of correlations have been explored. Entanglement has continued to be the most popular of these correlations owing to its inherent potential advantages in performing quantum computation and communication tasks horo-rmp (). More recently, however, there has been a rapidly growing interest in the study of correlations from a more direct measurement perspective modi-rmp (); celeri-rmp (), and several measures to quantify the same have been considered. Among these measures, quantum discord and classical correlation have been attracting much attention disc-imp1 (), and have lead to several interesting results disc-imp2 (); disc-imp3 ().

In this work, we undertake a comprehensive analysis of the problem of computation of correlations in the two-qubit system, especially the so-called -states xstates1 (); this class of states has come to be accorded a distinguished status in this regard xstates2 (). The problem of -states has already been considered in luo (); ali (); wang11 (); chen11 (); du-geom (); du12 (); huang13 () and that of more general two-qubit states in james11 (); adesso11 (); zambrini11 (); zambriniepl (); adesso11b (); nassajour13 (). The approach which we present here fully exploits the very geometric nature of the problem. In addition to being comprehensive, it helps to clarify and correct some issues in the literature regarding computation of correlations in -states. It may be emphasised that the geometric methods used here have been the basic tools of (classical) polarization optics for a very long time, and involve elementary constructs like Stokes vectors, Poincaré sphere, and Mueller matrix simon-mueller1 (); simon-mueller2 (); simon-mueller3 (); simon-mueller4 (). In this sense our approach to quantum discord and classical correlation is one inspired by classical polarization optics.

We assume, unless otherwise stated, that measurements are performed on subsystem . The expression for quantum discord is then given by zurek01 (); vedral01 ()

 D(^ρAB) =I(^ρAB)−C(^ρAB), I(^ρAB) =S(^ρA)+S(^ρB)−S(^ρAB), (1)

where denotes the mutual information which is supposed to capture the total correlation in the given bipartite state winter05 (). The second quantity is the maximum amount of classical correlation that one could extract as a result of measurements on subsystem . Now, classical correlation in a bipartite state is given by the expression vedral01 ()

 C(^ρAB)=maxΠB[S(^ρA)−∑jpjS(^ρAj)], (2)

where the probabilities are given by

 pj=Tr[(11A⊗ΠBj)^ρAB], (3)

the (normalized) state of system after measurement being given by

 ^ρAj=TrB[ΠBj^ρAB]pj. (4)

The set meets the defining conditions , and for all . That is, forms a POVM. The second term in the expression (2) for classical correlation is the (minimum, average) conditional entropy post measurement, and we may denote it by

 SAmin=minΠB∑jpjS(^ρAj), (5)

the minimum being taken over the set of all POVM’s. Then the expression for classical correlation simply reads as

 C(^ρAB)=S(^ρA)−SAmin, (6)

and, consequently, that for quantum discord as

 D(^ρAB)=S(^ρB)−S(^ρAB)+SAmin. (7)

Finally, we note that the first two terms of this expression for quantum discord are known as soon as the bipartite state is specified. Therefore the only quantity of computational interest is the conditional entropy of system post measurement (on B) : this alone involves an optimization. It is to the task of computing that the methods of classical polarization optics seem to be the most appropriate tools.

The content of the paper is organised as follows. We begin by indicating in Section II why we believe that the Mueller-Stokes formalism of classical polarization optics is the most appropriate tool for analyzing conditional states post measurement, and hence for computing quantum discord of a two-qubit state. The Mueller matrix associated with the density operator of an -state is presented in Section III, and the correlation ellipsoid of (normalized) conditional states post measurement associated with a two-qubit state (or its Mueller matrix) is analysed in Section IV. With the geometric tool of the correlation ellipsoid on hand, the problem of computation of the optimal mean conditional entropy is taken up in Sections V and VI. Our primary aim in Section V is to prove that the present optimization problem is one of convex optimization over an ellipse rather than over an ellipsoid, and hence optimization over a single variable. The actual computation of is comprehensively treated in Section VI, bringing out clearly all possible situations that could arise. With measurements assumed to be carried out on the B-side, the computational aspects which depend on the reduced state are clearly demarcated from those which do not. The correlation ellipsoid has an invariance group which is much larger than the group of local unitaries. The manner in which this larger invariance group helps the analysis is discussed in Section VII, including the manner in which it helps to connect the separability of a two-qubit -state directly to its correlation ellipsoid. Section VIII is devoted to a detailed comparison of our results with those of Ali, Rau and Alber ali (). -states of vanishing discord are fully enumerated in Section IX, and contrasted with earlier enumerations. Finally, -states for which the discord can be written down by inspection, with no need for optimization, are considered in Section X. This family is much larger than the -states treated in the well-known work of Luo luo ().

A comment may be in order before we turn to presentation of our analysis and results. While our treatment is geometrical in nature, the emphasis is on comprehensiveness. Thus, while many of our results are new, it is possible that some are known in scattered form in the works of earlier authors. For instance, while it is known from some earlier publications wang11 (); chen11 (); adesso11 (); zambrini11 (); zambriniepl (); du12 (); huang13 () that the main theorem of Ali, Rau, and Alber numerically fails for some -states, the present work seems to be the first to demonstrate that their very proof of the theorem itself is untenable in a fundamental manner. While we prove that the numerical failure applies to only a very tiny region of the space of -states, the failure of their proof would seem to apply not just to this tiny region but to all -states since the symmetry on which they base the proof of their theorem is a property of generic -states.

## Ii Mueller-Stokes formalism for two-qubit states

We begin with a brief indication as to why the Mueller-Stokes formalism of classical optics is possibly the most appropriate tool for handling quantum states post measurement. In classical polarization optics the state of a light beam is represented by a complex positive matrix called the polarization matrixneillbook (). The intensity of the beam is identified with , and so the matrix (normalized to unit trace) represents the actual state of polarization. The polarization matrix is thus analogous to the density matrix of a qubit, the only distinction being that the trace of the latter needs to assume unit value. Even this one little difference is gone when one deals with conditional quantum states post measurement : the probability of obtaining a conditional state becomes analogous to intensity of the classical context.

The Mueller-Stokes formalism itself arises from the following simple fact : any matrix can be invertibly associated with a four-vector , called the Stokes vector, through

 Φ=123∑k=0Skσk,  Sk=Tr(σkΦ). (8)

This representation is an immediate consequence of the fact that the Pauli triplet and , the unit matrix, form a complete orthonormal set of (hermitian) matrices.

Clearly, hermiticity of the polarization matrix is equivalent to reality of the associated four-vector and . Positivity of reads , corresponding, respectively, to the pair , . Thus positive matrices (or their Stokes vectors) are in one-to-one correspondence with points of the positive branch of the solid light cone. Unit trace (intensity) restriction corresponds to the section of this cone at unity along the ‘time’ axis, . The resulting three-dimensional unit ball is the more familiar Bloch (Poincaré) ball, whose surface or boundary representing pure states (of unit intensity) is often called the Bloch (Poincaré) sphere. The interior points correspond to mixed (partially polarized) states.

Optical systems which map Stokes vectors linearly into Stokes vectors have been of particular interest in polarization optics. Such a linear system is represented by a real matrix , the Mueller matrix simon-mueller1 (); simon-mueller2 (); simon-mueller3 (); simon-mueller4 () :

 M:Sin→Sout=MSin. (9)

It is evident that a (physical) Mueller matrix should necessarily map the positive solid light cone into itself. It needs to respect an additional subtle restriction, even in classical optics.

Remark 1 : The Mueller-Stokes formulation of classical polarization optics traditionally assumes plane waves. It would appear, within such a framework, one need not possibly place on a Mueller matrix any more demand than the requirement that it map Stokes vectors to Stokes vectors. However, the very possibility that the input (classical) light could have its polarization and spatial degrees of freedom intertwined in an inseparable manner, leads to the additional requirement that the Mueller matrix acting ‘locally’ on the polarization indices alone map such an entangled (classical) beam into a physical beam at the output. Interestingly, it is only recently that such an entanglement-based requirement has been established simon-mueller3 (); simon-mueller4 (), leading to a full characterization of Mueller matrices in classical polarization optics.

To see the connection between Mueller matrices and two-qubit states unfold naturally, use a single index rather than a pair of indices to label the computational basis two-qubit states in the familiar manner : . Now note that a two-qubit density operator can be expressed in two distinct ways :

 ^ρAB =3∑j,k=0ρjk|j⟩⟨k| =143∑a,b=0Mabσa⊗σ∗b, (10)

the second expression simply arising from the fact that the sixteen hermitian matrices form a complete orthonormal set of matrices. Hermiticity of operator is equivalent to reality of the matrix , but the same hermiticity is equivalent to being a hermitian matrix.

Remark 2 : It is clear from the defining equation (10) that the numerical entries of the two matrices thus associated with a given two-qubit state be related in an invertible linear manner. This linear relationship has been in use in polarization optics for a long time simon-mueller1 (); simon-mueller4 () and, for convenience, it is reproduced in explicit form in the Appendix.

Given a bipartite state , the reduced density operators of the subsystems are readily computed from the associated  :

 ^ρA =Tr[^ρAB]=123∑a=0Ma0σa, ^ρB =Tr[^ρAB]=123∑b=0M0bσ∗b. (11)

That is, the leading column and leading row of are precisely the Stokes vectors of reduced states respectively.

It is clear that a generic POVM element is of the form . We shall call the Stokes vector of the POVM element . Occasionally one finds it convenient to write it in the form with the ‘spatial’ -vector part highlighted. The Stokes vector corresponding to a rank-one element has components that satisfy the relation . Thus, rank-one elements are light-like and rank-two elements are strictly time-like. One recalls that similar considerations apply to the density operator of a qubit as well.

The (unnormalised) state operator post measurement (measurement element ) evaluates to

 ρAπj =TrB[^ρABΠBj] =18TrB[(3∑a,b=0Mabσa⊗σ∗b)(3∑k=0Skσ∗k)] =183∑a,b=03∑k=0MabSkσaTr(σ∗bσ∗k) =143∑a=0S′aσa, (12)

where we used in the last step.

Remark 3 : It may be emphasised, for clarity, that we use Stokes vectors to represent both measurement elements and states. For instance, Stokes vector in Eq. (12) stands for a measurement element on the B-side, whereas stands for (unnormalised) state of subsystem .

The Stokes vector of the resultant state in Eq. (12) is thus given by , which may be written in the suggestive form

 Sout=MSin. (13)

Comparison with (9) prompts one to call the Mueller matrix associated with two-qubit state . We repeat that the conditional state need not have unit trace, and so needs to be normalised when computing entropy post measurement. To this end, we write

 ρAπj =pj^ρπj pj=Sout02,  ^ρπj =12(11+(Sout0)−1Sout.σ). (14)

It is sometimes convenient to write the Mueller matrix associated with a given state in the block form

 M=(1ξTλΓ),  λ,ξ∈R3.

Then the input-output relation (13) reads

 Sout0=Sin0+ξ⋅Sin,   Sout =Sin0λ+ΓSin, (15)

showing in particular that the probability of the conditional state on the A-side depends on the POVM element precisely through .

Remark 4 : The linear relationship between two-qubit density operators (states) and Mueller matrices (single qubit maps) we have developed in this Section can be usefully viewed as an instance of the Choi-Jamiokowski isomorphism cj ().

Remark 5 : We have chosen measurements to be made on the B qubit. Had we instead chosen to compute correlations by performing measurements on subsystem then, by similar considerations as detailed above, we would have found playing the role of the Mueller matrix .

## Iii X-states and their Mueller matrices

-states are states whose density matrix has non-vanishing entries only along the diagonal and the anti-diagonal. That is, the numerical matrix has the ‘shape’ of . A general -state can thus be written, to begin with, as

 ρX=⎛⎜ ⎜ ⎜ ⎜⎝ρ0000ρ03eiϕ20ρ11ρ12eiϕ100ρ12e−iϕ1ρ220ρ03e−iϕ200ρ33⎞⎟ ⎟ ⎟ ⎟⎠, (16)

where the ’s are all real nonnegative. One can get rid of the phases (of the off-diagonal elements) by a suitable local unitary transformation . This is not only possible, but also desirable because the quantities of interest, namely entanglement, mutual information, quantum discord and classical correlation, are all invariant under local unitary transformations. Since it is unlikely to be profitable to carry around a baggage of irrelevant parameters, we shall indeed shed by taking to its canonical form . We have

 ρX→ρcanX=UA⊗UBρXU†A⊗U†B, (17)

where

 ρcanX =⎛⎜ ⎜ ⎜ ⎜⎝ρ0000ρ030ρ11ρ1200ρ12ρ220ρ0300ρ33⎞⎟ ⎟ ⎟ ⎟⎠; UA =diag(e−i(2ϕ1+ϕ2)/4,eiϕ2/4), UB =diag(ei(2ϕ1−ϕ2)/4,eiϕ2/4). (18)

Remark 6 : We wish to clarify that -states thus constitute, in the canonical form, a (real) 5-parameter family, three diagonal parameters () and two off-diagonal parameters; it can be lifted, using local unitaries which have three parameters each, to a -parameter subset in the -parameter state space (or generalized Bloch sphere) of two-qubit states : they are all local unitary equivalent to the conventional -states, though they may no more have ‘shape’ .

With this canonical form, it is clear that the Mueller matrix for the generic -state has the form

 M=⎛⎜ ⎜ ⎜⎝100m030m110000m220m3000m33⎞⎟ ⎟ ⎟⎠, (19)

where

 m11 =2(ρ03+ρ12),  m22=2(ρ03−ρ12), m03 =ρ00+ρ22−(ρ11+ρ33), m33 =ρ00+ρ33−(ρ11+ρ22), m30 =ρ00+ρ11−(ρ22+ρ33), (20)

as can be read off from the defining equation (10) or from the relation in the Appendix. We note that the Mueller matrix of an -state has a ‘sub-X’ form : the only nonvanishing off-diagonal entries are and (). In our computation later we will sometimes need the inverse relations

 ρ00 =14(m00+m03+m30+m33), ρ11 =14(m00−m03+m30−m33), ρ22 =14(m00+m03−m30−m33), ρ33 =14(m00−m03−m30+m33), ρ03 =14(m11+m22),  ρ12=14(m11−m22). (21)

The positivity properties of , namely , transcribes to the following conditions on the entries of its Mueller matrix :

 (1+m33)2 −(m30+m03)2≥(m11+m22)2 (22) (1−m33)2 −(m30−m03)2≥(m11−m22)2. (23)

Remark 7 : As noted earlier the requirements (22), (23) on Mueller matrix (19) in the classical polarization optics context was established for the first time in Refs. simon-mueller3 (); simon-mueller4 (). These correspond to complete positivity requirement on considered as a positive map (map which images the solid light cone into itself), and turns out to be equivalent to positivity of the corresponding two-qubit density operator.

By virtue of the direct-sum block structure of -state density matrix, one can readily write down its (real) eigenvectors. We choose the following order for definiteness :

 |ψ0⟩ =cα|00⟩+sα|11⟩,  |ψ1⟩=cβ|01⟩+sβ|10⟩, |ψ2⟩ =−sβ|01⟩+cβ|10⟩,|ψ3⟩=−sα|00⟩+cα|11⟩, (24)

where denote respectively and . And (dropping the superscript ‘can’) we have the spectral resolution

 ^ρX=3∑j=0λj|ψj⟩⟨ψj|, (25)
 cα =√1+ν12, cβ=√1+ν22, ν1 =ρ00−ρ33√4ρ203+(ρ00−ρ33)2=m30+m03√(m11+m22)2+(m30+m03)2, ν2 =ρ11−ρ22√4ρ212+(ρ11−ρ22)2=m30−m03√(m11−m22)2+(m30−m03)2; λ0or3 =ρ00+ρ332±√(ρ00−ρ33)2+4ρ2032 =1+m334±√(m11+m22)2+(m30+m03)24, λ1or2 =ρ11+ρ222±√(ρ11−ρ22)2+4ρ2122 =1−m334±√(m11−m22)2+(m30−m03)24. (26)

While computation of will have to wait for a detailed consideration of the manifold of conditional states of , the other entropic quantities can be evaluated right away. Given a qubit state specified by Stokes vector , it is clear that its von Neumann entropy equals

 S2(r)=−[1+r2]ℓog2[1+r2]−[1−r2]ℓog2[1−r2], (27)

where is the norm of the three vector , or the distance of from the origin of the Bloch ball. Thus from Eq. (11) we have

 S(^ρA) =S2(|m30|), S(^ρB)=S2(|m03|), S(^ρAB) ≡S2({λj})=3∑j=0−λjℓog2(λj), (28)

where , are the eigenvalues of the bipartite state given in Eq. (26). The mutual information thus assumes the value

 I(^ρAB)=S2(|m30|)+S2(|m03|)−S2({λj}). (29)

## Iv Correlation ellipsoid : Manifold of conditional states

We have seen that the state of subsystem resulting from measurement of any POVM element on the B-side of is the Stokes vector resulting from the action of the associated Mueller matrix on the Stokes vector of the POVM element. In the case of rank-one measurement elements, the ‘input’ Stokes vectors correspond to light-like points on the (surface of the) Bloch ball. Denoting the POVM elements as , , we ask for the collection of corresponding normalized conditional states. By Eq. (13) we have

 Sout =MSin=⎛⎜ ⎜ ⎜⎝1+m03zm11xm22ym30+m33z⎞⎟ ⎟ ⎟⎠→⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1m11x1+m03zm22y1+m03zm30+m33z1+m03z⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (30)

It is clear that, for , whenever and the input is not in the x-y plane of the Poincaré sphere. It can be shown that the sphere at the ‘input’ is mapped to the ellipsoid

 x2a2x+y2a2y+(z−zc)2a2z=1 (31)

of normalized states at the output, the parameters of the ellipsoid being fully determined by the entries of  :

 ax =|m11|√1−m203,   ay=|m22|√1−m203, az =|m33−m03m30|1−m203,   zc=m30−m03m331−m203. (32)

Remark 8 : This ellipsoid of all possible (normalized) conditional states associated with a two-qubit state is sometimes known as the steering ellipsoid du-geom (); du12 (); rudolph13 (). It degenerates into a single point if and only if the state is a product or uncorrelated state. It captures in a geometric manner correlations in the two-qubit state under consideration, and correlation is the object of focus in the present work. For these reasons, we prefer to call it the correlation ellipsoid associated with the given two-qubit state. While measurement elements are mapped to points of the ellipsoid, measurement elements for all and fixed are mapped to one and the same point of the correlation ellipsoid. Thus, in the general case, each point of the ellipsoid corresponds to a ‘ray’ of measurement elements. In the degenerate case wherein the ellipsoid becomes a disc or line segment or a single point and only in that case, do several rays map to the same point.

The x-z section of the correlation ellipsoid is pictorially depicted in Fig. 1. It is clear that the geometry of the ellipsoid is determined by the four parameters and could be assumed nonnegative without loss of generality. The fifth parameter specifying the z-coordinate of the image of the maximally mixed state as measurement element on the B side, is not part of this geometry. It is clear that corresponds to .

Having thus considered the passage from a two-qubit -state to its correlation ellipsoid, we may raise the converse issue of going from the correlation ellipsoid to the associated -state. To do this, however, we need the parameter as an input in addition to the ellipsoid itself. Further, change of the signature of does not affect the ellipsoid in any manner, but changes the states and correspondingly the signature of . Thus, the signature of needs to be recorded as an additional binary parameter. It can be easily seen that the nonnegative along with and fully reconstruct the -state in its canonical form (18), (19) [see Remark 16]. Using local unitary freedom we can render and nonnegative so that ; can assume either signature. It turns out to be convenient to denote by the collection of all Mueller matrices with and by those with . The intersection corresponds to Mueller matrices for which , a measure zero subset. Further, in our analysis to follow we assume, without loss of generality,

 ax≥ay, i.e., m11≥|m22|. (33)

Remark 9 : Every two-qubit state has associated with it a unique correlation ellipsoid of (normalized) conditional states. An ellipsoid centered at the origin needs six parameters for its description : three for the sizes of the principal axes and three for the orientation of the ellipsoid as a rigid body in . For a generic (i.e., not necessarily ) state , the centre can be shifted from the origin to vectorial location , thus accounting for three parameters, and can be located at anywhere inside the ellipsoid, thus accounting for another three. The three-parameter local unitary freedom on the B-side, which has no effect whatsoever on the geometry of the ellipsoid (but determines which points of the input Poincaré sphere go to which points on the surface of the ellipsoid) accounts for the final three parameters, adding to a total of . For -states the shift of from the origin needs to be along one of the principal directions and is constrained to be located on this very principal axis. In other words, and become one-dimensional rather than three-dimensional variables rendering -states a 11-parameter subfamily of the 15-parameter state space. Thus -states are distinguished by the fact that , , and the origin are collinear with one of the principal axes of the ellipsoid. This geometric rendering pays no special respect to the shape , but is manifestly invariant under local unitaries as against the characterization in terms of ‘shape’ of the matrix in the computation basis. Since the latter (conventional) characterization is not even invariant under local unitaries, we are tempted to a strong appeal to the community in favour of our invariant geometric characterization of -states.

## V Optimal measurement

In this Section we take up the central part of the present work which is to develop a provably optimal scheme for computation of the quantum discord for any -state of a two-qubit system. Our treatment is both comprehensive and self-contained and, moreover, it is geometric in flavour. We begin by exploiting symmetry to show, without loss of generality, that the problem itself is one of optimization in just a single variable. The analysis is entirely based on the output or correlation ellipsoid associated with a two-qubit state , and we continue to assume that measurements are carried out on the B-side.

The single-variable function under reference will be seen, on optimization, to divide the manifold of possible correlation ellipsoids into two subfamilies. For one subfamily the optimal measurement or POVM will be shown to be a von Neumann measurement along either x or z, independent of the location (inside the ellipsoid) of , the image of the maximally mixed input. For the other subfamily, the optimal POVM will turn out to be either a von Neumann measurement along x or a three-element POVM, depending on the actual location of in the ellipsoid. There exists no -state for which the optimal measurement requires a four-element POVM, neither does there exist an -state for which the optimal POVM is von Neumann in a direction which is neither along x nor z.

For the special case of the centre of the ellipsoid coinciding with the origin of the Poincaré sphere (), it will be shown that the optimal measurement is always a von Neumann measurement along x or z, irrespective of the location of in the ellipsoid. While this result may look analogous to the simple case of Bell mixtures earlier treated by Luo luo (), it should be borne in mind that these centred -states form a much larger family than the family of Bell mixtures, for in the Luo scenario necessarily coincides with and with the origin, but we place no such restriction of coincidence. Stated differently, in our case of centered ellipsoids is an independent variable in addition to . We shall return to the case of centered ellipsoids in Section X.

As we now turn to the analysis itself it is useful to record this : the popular result that the optimal POVM requires no more than four elements plays a priori no particular role of help in our analysis; it is for this reason that we shall have no occasion in our analysis to appeal to this important theorem zaraket04 (); ariano05 ().

Proposition 1 : The optimal POVM needs to comprise rank-one elements.
Proof : This fact is nearly obvious, and equally obvious is its proof. Suppose is a rank-two element of an optimal POVM and the associated conditional state of subsystem . Write as a positive (convex) sum of rank-one elements and let be the conditional states corresponding respectively to . It is then clear that , for some . Concavity of the entropy function immediately implies , in turn implying through (5) that the POVM under consideration could not have been optimal, thus completing the proof. It is clear from the nature of the proof that this fact applies to all states and not just to -states, and to all Hilbert space dimensions and not just .

Remark 10 : Since a rank-one POVM element is just a point on (the surface of) the Bloch (Poincaré) sphere , a four element rank-one POVM is a quadruple of points on , with associated probabilities . The POVM condition demands that we have to solve the pair

 p1+p3 +p3+p4=2, ∑jpjS(j)=0. (34)

Once four points on are chosen, the ‘probabilities’ are not independent. To see this, consider the tetrahedron for which are the vertices. If this tetrahedron does not contain the origin, then has no solution with nonnegative . If it contains the origin, then there exits a solution and the solution is ‘essentially’ unique by Caratheodory theorem.

The condition comes into play in the following manner. Suppose we have a solution to . It is clear that , , with no change in ’s, will also be a solution for any (-independent) . It is this freedom in choosing the scale parameter that gets frozen by the condition , rendering the association between tetrahedra and solutions of the pair (34) indeed unique.

We thus arrive at a geometric understanding of the manifold of all (rank-one) four-element POVM’s (even though one would need such POVM’s only when one goes beyond -states). This is precisely the manifold of all tetrahedra with vertices on , and containing the centre in the interior of . We are not considering four-element POVM’s whose are coplanar with the origin of , because they are of no use as optimal measurements. It is clear that three element rank-one POVM’s are similarly characterized, again by the Caratheodory theorem, by triplets of points on coplanar with the origin of , with the requirement that the triangle generated by the triplet contains the origin in the interior. Further, it is trivially seen in this manner that 2-element rank-one POVM’s are von Neumann measurements determined by pairs of antipodal ’s on , i.e., by ‘diameters’ of .

The correlation ellipsoid of an -state (as a subset of the Poincaré sphere) has a symmetry generated by reflections respectively about the x-z and y-z planes. We shall now use the product of these two reflections—a -rotation or inversion about the z-axis—to simplify, without loss of generality, our problem of optimization.

Proposition 2 : All elements of the optimal POVM have to necessarily correspond to (light-like) Stokes vectors of the form , i.e., the measurement elements are constrained to the x-z plane.
Proof : Suppose is an optimal POVM of rank-one elements (we are placing no restriction on the cardinality of , but rather expect it to unfold naturally from the analysis to follow). And let be the corresponding (normalized) conditional states, these being points on the boundary of the correlation ellipsoid. Let and represent, respectively, the images of , under -rotation about the z-axis (of the input Poincaré sphere and of the correlation ellipsoid) : , . It follows from symmetry that too is an optimal POVM. And so is also , where we have used the decorated symbol rather than the set union symbol to distinguish from simple union of sets : if happens to be an element of , then for this element, and in that case this should be ‘included’ in not once but twice (equivalently its ‘weight’ needs to be doubled). The same consideration holds if includes any and .

Our supposed to be optimal POVM can thus be assumed to comprise pairs of elements related by inversion about the z-axis. Let us consider the associated pair of conditional states on the (surface of the) correlation ellipsoid. They have identical z-coordinate . The section of the ellipsoid (parallel to the x-y or equatorial plane) at is an ellipse, with major axis along (recall (33) wherein we have assumed, without loss of generality, ), and and are on opposite ends of a line segment through the centre of the ellipse. Let us assume that this line segment is not the major axis of the ellipse . That is, we assume , are not in the x-z plane.

Now slide (only) this pair along the ellipse smoothly, keeping them at equal and opposite distance from the z-axis until both reach opposite ends of the major axis of the ellipse, the x-z plane. It is clear that during this process of sliding both , recede away from the centre of the ellipse and hence away from the centre of the Poincaré sphere itself. As a result decreases, thus improving the value of in (5). This would have proved that the POVM is not optimal, unless our assumption that , are not in the x-z plane is false. This completes proof of the proposition.

This preparation immediately leads to the following important result which forms the basis for our further analysis.

###### Theorem 1

: The problem of computing quantum discord for -states is a problem of convex optimization on a plane, and optimization over a single variable.

Proof : We have just proved that elements of the optimal POVM come, in view of the symmetry of -states, in pairs of Stokes vectors with . The corresponding conditional states come in pairs , . The two states of such a pair of conditional states are at the same distance

 r(zj)=√z2j+a2x−(zj−zc)2a2x/a2z (35)

from the origin of the Poincaré sphere, and hence they have the same von Neumann entropy

 f(zj) =S2(r(zj)), S2(r) =−1+r2ℓog21+r2−1−r2ℓog21−r2. (36)

Further, continuing to assume without loss of generality , our convex optimization is not over the three-dimensional ellipsoid, but effectively a planar problem over the x-z elliptic section of the correlation ellipsoid (Proposition 2), and hence the optimal POVM cannot have more that three elements. Thus, the (Stokes vectors of the) optimal POVM elements on the B-side necessarily have the form,

 Π(3)θ ={2p0(θ)(1,0,0,1)T,2p1(θ)(1,±sinθ,0,−cosθ)T}, p0(θ) =cosθ1+cosθ,  p1(θ)=1cosθ, 0≤θ≤π/2. (37)

The optimization itself is thus over the single variable .

Remark 11 : It is clear that and correspond respectively to von Neumann measurement along z and x, and no other von Neumann measurement gets included in . Every in the open interval corresponds to a genuine three-element POVM. The symmetry considerations above do allow also three-element POVM’s of the form

 ˜Π(3)θ ={2p0(θ)(1,0,0,−1)T,2p1(θ)(1,±sinθ,0,cosθ)T},  0≤θ≤π/2, (38)

but such POVM’s lead to local maximum rather than minimum for , and hence are of no value to us.

## Vi Computation of SAmin

A schematic diagram of the 3-element POVM of Eq. (37) is shown in Fig. 2. The Bloch vectors of the corresponding conditional states at the output are found to be of the form

 (0,0,zc+az)T, (x(z),0,z)T,(−x(z),0,z)T, x(z)=axaz(a2z−(z−zc)2)1/2. (39)

For these states denoted in Fig. 2 the weights should be chosen to realize as convex sum the state (the image of the maximally mixed input) whose Bloch vector is . von Neumann measurements along the z or x-axis correspond respectively to or .

Using Eqs. (35), (36), the expression for is thus given by

 SA(z)=p1(z)f(zc+az)+p2(z)f(z), p1(z)=zI−zzc+az−z,  p2(z)=zc+az−zIzc+az−z. (40)

The minimization of with respect to the single variable should give . It may be noted in passing that, for a given or , the three-element POVM parametrized by makes no sense in the present context for (since ought to be ).

For clarity of presentation, we begin by considering a specific example . To begin with, the relevant interval for the variable in this case is , and we shall examine the situation as we vary for fixed . The behaviour of for this example is depicted in Fig. 3, wherein each curve in the plane corresponds to a chosen value of , and the value of increases as we go down Fig. 3. For values of , for some to be detailed later, is seen to be a monotone increasing function of , and so its minimum