Progress on Quantum Discord of Two Qubit states: Optimization and Upper Bound
S. Javad Akhtarshenas, Hamidreza Mohammadi, Fahimeh S. Mousavi, Vahid Nassajpour
aa r X i v : . [ qu a n t - ph ] A p r Progress on Quantum Discord of Two-Qubit States: Optimization and Upper Bound
S. Javad Akhtarshenas,
1, 2, 3
Hamidreza Mohammadi,
1, 2
Fahimeh S. Mousavi, and Vahid Nassajpour Department of Physics, University of Isfahan, Isfahan, Iran Quantum Optics Group, University of Isfahan, Isfahan, Iran Department of Physics, Ferdowsi University of Mashhad, Mashhad, Iran
Calculation of the quantum discord requires to find the minimum of the quantum conditionalentropy S ( ρ AB |{ Π Bk } ) over all measurements on the subsystem B . In this paper, we provide asimple relation for the conditional entropy as the difference of two Shannon entropies. The relationis suitable for calculation of the quantum discord in the sense that it can be used to obtain thequantum discord for some classes of two-qubit states such as Bell-diagonal states and a three-parameter subclass of X states, without the need for minimization. We also present an analyticalprocedure of optimization and obtain conditions under which the quantum conditional entropy ofa general two-qubit state is stationary. The presented relation is also used to find a tight upperbound on the quantum discord. PACS numbers: 03.67.-a, 03.65.Ta, 03.65.Ud
I. INTRODUCTION
Entanglement is the specific feature of quantum sys-tems which reveals that complete information aboutparts of a composite system does not include completeinformation of the whole system [1, 2]. However, thisis not the only weird character of a quantum system.For instance, the collapse of one part of a non-entangledbipartite quantum system after a measurement on theother part is another feature unique to quantum systems.The quantity that captures this feature is the quantumdiscord [3, 4]. The key idea of the concept of quantumdiscord is the superposition principle and the vanishingof discord shown to be a criterion for quantum to classicaltransition [3]. Furthermore, quantum discord has a sim-ple interpretation in thermodynamics and has been usedin analyzing the power of a quantum Maxwell’s demon[5]. It has also been employed in the study of pure quan-tum states as a resource and performance of deterministicquantum computation with one pure qubit [6]. The au-thors of Refs. [7, 8] showed that non-classical correlationsother than entanglement can be responsible for the quan-tum computational efficiency of deterministic quantumcomputation with one pure qubit [6] and brought this ob-scured correlation measure to the spot light zone. Afterthis discovery, quantum discord becomes one of the mostfrequent topics of researches in the field of quantum in-formation theory. Indeed, quantum discord is the differ-ence between two classically equivalent definitions of mu-tual information in the quantum mechanics language. Inmathematical sense, discord could be obtained by elim-inating the whole of classical correlation from the totalcorrelation measured by mutual information, by meansof the most destructive measurement on the one partyof the system. Mutual information of a bipartite systemcan be written as I ( ρ AB ) = S ( ρ A ) + S ( ρ B ) − S ( ρ AB ) , (1)where ρ A and ρ B refer to the reduced density matri-ces of the subsystems A and B , respectively, ρ AB is the density matrix of the system as the whole, and S ( ρ ) = − Tr( ρ log ρ ) is the Von Neumann entropy. Theclassical correlation between the parts of a bipartite sys-tem can be obtained by use of the measurement-baseconditional density operator and can be written as [3] C B ( ρ AB ) = sup { Π Bk } { S ( ρ A ) − S ( ρ A |{ Π Bk } ) } . (2)Here the maximum is taken over all projective measure-ment { Π Bk } on the subsystem B [3], and S ( ρ A |{ Π Bk } ) = P k p k S ( ρ Ak ) is the conditional entropy of the subsys-tem A , with ρ Ak = Tr B (( I A ⊗ Π Bk ) ρ AB ( I A ⊗ Π Bk )) /p k as the post-measurement state of the subsystem A and p k = Tr( ρ AB ( I A ⊗ Π Bk )) being the probability of the k -thoutcome. The maximum performed in Eq. (2) can betaken also over all the positive operator valued measures(POVM) [4] and these two definitions give in general in-equivalent results. Accordingly, discord can be calculatedas follows D B ( ρ AB ) = I ( ρ AB ) − C B ( ρ AB ) . (3)However, one can swap the role of the subsystems A and B to obtain D A ( ρ AB ), which is not equal to D B ( ρ AB )in general. In this paper we only consider D B ( ρ AB ) andhence ignore the subscript B in the following.The optimization procedure involved in the calculationof quantum discord prevents one to write an analyticalexpression for quantum discord even for simple two-qubitsystems. Quantum discord is analytically computed onlyfor a few families of states including the Bell-diagonalstates [9, 10], two-qubit X states [11, 12], two-qubit rank-2 states [13], a class of rank-2 states of 4 ⊗ S ( ρ AB |{ Π Bk } ) as the difference of two Shannon entropies.This form of the conditional entropy enables one to calcu-late its minimum for some classes of two-qubit states suchas Bell-diagonal states [9] and a three-parameter sub-class of X states [11, 12], without any minimization pro-cedure. Although, the quantum discord of these statesis already obtained analytically, but the presented formfor the conditional entropy enables one to obtain the pre-vious results in a much simpler manner. An analyticalprogress in the minimization of the conditional entropyof a general two-qubit state is also presented. Our algo-rithm presents a necessary and sufficient condition for ameasurement to be the stationary measurement for theconditional entropy. The presented condition is, to thebest of our knowledg, more efficient relative to the earlierpresented optimization algorithms. Moreover, we obtaina computable tight upper bound on the quantum discordof an arbitrary two-qubit state. We also present suffi-cient conditions under which the upper bound is tigh,and exemplify this bound for a two-parameter class ofstates and show that the bound may be tight even in theabsence of such sufficient conditions.The paper is organized as follows. In section II, weconsider a general two-qubit system and present a sim-ple relation for the quantum conditional entropy. In thissection, we also evaluate the quantum discord for someclasses of states. In section III, we present a tight upperbound on the discord of a general two-qubit state. Sec-tion IV is devoted to the optimization procedure. An an-alytical conditions under which the conditional entropyis stationary is presented in this section. The paper isconcluded in section V with a brief discussion. II. CONDITIONAL ENTROPY
A general two-qubit state can be written in the Hilbert-Schmidt representation as ρ AB = 14 I ⊗ I + ~x · σ ⊗ I + I ⊗ ~y · σ + X i,j =1 t ij σ i ⊗ σ j . (4)Here I stands for the identity operator, { σ i } i =1 are thestandard Pauli matrices, ~x and ~y are coherence vectorsof the subsystems A and B , respectively, and T = ( t ij ) isthe correlation matrix. Therefore, to each state ρ AB weassociate the triple { ~x, ~y, T } . Since quantum correlationsare invariant under local unitary transformation, i.e. un-der transformations of the form ( U ⊗ U ) ρ AB ( U † ⊗ U † )with U , U ∈ SU (2), we can, without loss of generality,restrict our considerations to some representative class ofstates described by less number of parameters [19]. Un-der such transformations, the triple { ~x, ~y, T } transformsas ~x → O ~x, ~y → O ~y, T → O T O t2 , (5) where O i ’s corresponds to U i ’s via U i ( ~a · ~σ ) U † i = ( O i ~a ) · ~σ [19]. In view of this, any state of the two-qubit systemcan be written as ( U ⊗ U ) ρ AB ( U † ⊗ U † ), where ρ AB be-longs to the representative class. In the following we willconsider a representative class such that T is diagonal,namely T = diag { t , t , t } . Concerning this represen-tative class, a general state of two-qubit system can beparameterized by nine real parameters ~x = ( x , x , x ) t , ~y = ( y , y , y ) t , and T = diag { t , t , t } , where t de-notes transposition. Accordingly, in the computationalbasis {| i , | i , | i , | i} , a general state ρ AB of thisrepresentative class takes the following form ρ AB = 14 ρ y − iy x − ix t − t y + iy ρ t + t x − ix x + ix t + t ρ y − iy t − t x + ix y + iy ρ , (6)where ρ = 1 + x + y + t , ρ = 1 + x − y − t ,ρ = 1 − x + y − t , ρ = 1 − x − y + t . Now let us turn our attention on the von Neumann mea-surement on the qubit B. A general such measurementcan be written as Π Bk = U | k ih k | U † , (7)where {| k ih k |} k =0 is the von Neumann projection opera-tors in the standard basis of the qubit B , and U ∈ SU (2).An arbitrary element of SU (2) can be factorized as [20] U = Ω Ω , (8)with Ω and Ω defined byΩ = (cid:18) cos θ − e − iφ sin θ e iφ sin θ cos θ (cid:19) , Ω = (cid:18) e iη/
00 e − iη/ (cid:19) , (9)for 0 ≤ θ ≤ π , 0 ≤ φ ≤ π , and 0 ≤ η ≤ π . Therefor,we get Π Bk = Ω | k ih k | Ω † = | σ · ˆ n k ih σ · ˆ n k | where ˆ n = − ˆ n = ˆ n , with ˆ n = (sin θ cos φ, sin θ sin φ, cos θ ) t . This,explicitly, shows that only two independent parameters θ and φ are needed to characterize a general local vonNeumann measurement on the two-qubit systems. Wecan also write these orthogonal projections in the Blochrepresentation asΠ Bk = 12 ( I + ˆ n k · σ ) , k = 0 , . (10)Therefore ˆ n and ˆ n are coherence vectors of Π B and Π B ,respectively. For further use, we calculate the expressionΠ Bk σ j Π Bk for k = 0 , j = 1 , ,
3, i.e.Π Bk σ j Π Bk = | σ · ˆ n k ih σ · ˆ n k | σ j | σ · ˆ n k ih σ · ˆ n k | = Tr (cid:0) Π Bk σ j (cid:1) Π Bk = (ˆ n k ) j Π Bk . (11)Now we are in the position to perform the von Neumannmeasurement { Π Bk } k =0 on the qubit B . This transformsthe total state ρ AB to the ensemble { p k , ρ ABk } k =0 suchthat ρ ABk = 1 p k ( I ⊗ Π Bk ) ρ AB ( I ⊗ Π Bk ) , (12)with p k = Tr[( I ⊗ Π Bk ) ρ AB ( I ⊗ Π Bk )]. By using Eqs. (4)and (10) in (12) and invoking relation (11) we get ρ ABk = ρ Ak ⊗ Π Bk , (13)where ρ Ak = 12 (cid:16) I + ~ ˜ x k · σ (cid:17) , (14)is the post-measurement state of the qubit A , associatedto the measurement result k with the corresponding prob-ability p k = 12 (cid:0) ~y t ˆ n k (cid:1) . (15)In Eq. (14), the post-measurement coherence vector ~ ˜ x k is defined by ~ ˜ x k = ~x + T ˆ n k ~y t ˆ n k . (16)The quantum conditional entropy with respect to theabove measurement is defined by S ( ρ A |{ Π Bk } ) = p S ( ρ A |{ Π Bk } ) + p S ( ρ A |{ Π Bk } ) . (17)Now using12 (cid:16) ± | ~ ˜ x k | (cid:17) = 14 p k (2 p k ± | ~x + T ˆ n k | ) , (18)as the eigenvalues of ρ Ak , for k = 1 ,
2, and after somecalculations we arrive at the following observation for theconditional entropy.
Observation 1.
Conditional entropy can be written as S ( ρ A |{ Π Bk } ) = h ( ~w ) − h ( p ) , (19) where above, and throughout this paper, h ( x ) denotesthe binary Shannon entropy [21] defined by h ( x ) = − x log x − (1 − x ) log (1 − x ) , (20) and h m ( q , · · · , q m ) = − P mi =1 q i log q i is the Shannonentropy of the probabilities { q , · · · , q m } . In particular, h ( ~w ) = − P i =1 w i log w i is the Shannon entropy of theprobabilities w , = 2 p ± | ~x + T ˆ n | , w , = 2 p ± | ~x − T ˆ n | . (21)Note that under the transformation ˆ n → − ˆ n , corre-sponding to θ → π − θ and φ → φ ± π , the probabili-ties (15) and (21) transform as p ↔ p , w ↔ w and w ↔ w , leaving therefore the conditional entropy in-variant. On the other hand, if we perform local unitarytransformation (5) on the density matrix, the probabil-ities (15) and (21), and hence the conditional entropydo not change provided we perform the transformationˆ n → O ˆ n . This implies that if we find ˆ n ∗ as the optimalmeasurement for a given state ρ AB of the representa-tive class, one can obtain the optimal one for any state˜ ρ AB = ( U ⊗ U ) ρ AB ( U † ⊗ U † ), just by the transforma-tion ˆ n ∗ → O ˆ n ∗ .Now the aim is to minimize the above conditional en-tropy. Before we give a general procedure for optimiza-tion, we give below some special classes of states for whichthe quantum discord can be evaluated without the needfor any optimization. Quantum Discord of States with T t ~x = 0 and ~y = 0 . Let us consider a three-parameter class of states such that ~y = 0 and ~x belongs to the kernel of T t , i.e. T t ~x = 0.For this class of states we get | x + T ˆ n | = | x − T ˆ n | = p x + ˆ n t T t T ˆ n, (22)and therefore p = p = 12 , w , = w , = 14 (1 ± | ~x + T ˆ n | ) . (23)In this case we get h ( ~w ) = h (cid:18) | x + T ˆ n | (cid:19) + 1 , h ( p ) = 1 , (24)and therefore Eq. (19) reduces to S ( ρ A |{ ˆ n } ) = h (cid:18) | x + T ˆ n | (cid:19) . (25)Clearly, the minimum of the above equation occurs when-ever | x + T ˆ n | = √ x + ˆ n t T t T ˆ n takes its maximum value.This happens when ˆ n is an eigenvector of T t T corre-sponding to the largest eigenvalue t , thereforemin S ( ρ A |{ ˆ n } ) = h p x + t ! . (26)For these states quantum discord is D ( ρ AB ) = 1 − h ( µ , µ , µ , µ ) + h p x + t ! , where { µ i } i =1 are eigenvalues of ρ AB .Note that if we concern the representative class ofstates (6) for which T = diag { t , t , t } , then condition T t ~x = 0 requires that t i x i = 0 for i = 1 , ,
3. Hence if wetake, without loss of generality, | t | ≥ | t | ≥ | t | ≥ ~y = 0, then the states corresponding to this class canbe obtained from the general form of Eq. (6) as:(i) States with x = x = x = 0 .— This corresponds tothe Bell-diagonal states. In this case discord reads as D ( ρ AB ) = 1 − h ( µ , µ , µ , µ ) + h (cid:18) | t | (cid:19) , with { µ i } i =1 as eigenvalues of ρ AB given by µ , = 14 (1 ± t ± t − t ) , µ , = 14 (1 ± t ∓ t + t ) . This is in agreement with the result obtained by Luo in[9] (see also [10]).(ii)
States with x = x = t = 0 .— This corresponds toa three-parameter subclass of the so-called X states. Inthis case, the discord is obtained as D ( ρ AB ) = 1 − h ( µ , µ , µ , µ ) + h p t + x ! , where { µ i } i =1 are eigenvalues of ρ AB given by µ , = 14 (cid:18) ± q ( t + t ) + x (cid:19) ,µ , = 14 (cid:18) ± q ( t − t ) + x (cid:19) . (iii) States with x = t = t = 0 .— This corresponds toa three-parameter subclass of the zero-discord states.(iv)
States with t = t = t = 0 .— This also correspondsto a three-parameter subclass of the zero-discord states.
III. TIGHT UPPER BOUND OF QUANTUMDISCORD
Interestingly, the above examples motivate us to in-troduce an upper bound for the quantum discord. Let R be the subspace of R spanned by T t ~x and ~y , i.e. R = span { T t ~x, ~y } , and let R ⊥ denotes the orthogonalcomplement of R , i.e. the set of all vectors in R thatare orthogonal to every element of R . Hence we have R + R ⊥ = R . Theorem 1.
The conditional entropy (19) is boundedfrom above as min { Π Bk } S ( ρ A |{ Π Bk } ) ≤ h p x + t ! , (27) where x = | ~x | , and t = max ˆ e ∈R ⊥ ˆ e t0 T t T ˆ e . (28) Accordingly, the classical correlation and the quantumdiscord have the following lower and upper bounds, re-spectively C ( ρ AB ) ≥ S ( ρ A ) − h p x + t ! , (29) Q ( ρ AB ) ≤ S ( ρ B ) − S ( ρ AB ) + h p x + t ! . (30) Proof.
Let us concern about all measurement vectors ˆ e living in R ⊥ , i.e. ~y t ˆ e = 0, ( T t ~x ) t ˆ e = 0; then | x + T ˆ e | = | x − T ˆ e | = q x + ˆ e t0 T t T ˆ e , (31)and p = p = 12 , w , = w , = 14 (1 ± | ~x + T ˆ e | ) . (32)Thereforemin ˆ n ∈ R S ( ρ A |{ ˆ n } ) ≤ min ˆ e ∈R ⊥ S ( ρ A |{ ˆ e } )= min ˆ e ∈R ⊥ h p x + ˆ e t0 T t T ˆ e ! = h p x + t ! , (33)where t = max ˆ e t0 T t T ˆ e with the maximum taken overall unit vectors ˆ e ∈ R ⊥ . Evidently, if two vectors T t ~x and ~y be nonzero and linearly independent thendim R ⊥ = 1, so that the unit vector ˆ e ∈ R ⊥ is unique.Otherwise dim R ⊥ >
1, so we can choose ˆ e ∈ R ⊥ suchthat t = max ˆ e ∈R ⊥ ˆ e t0 T t T ˆ e , giving a tighter bound. Thiscompletes the proof of (27). Using Eq. (27) in Eqs. (2)and (3), we obtain the desired bounds (29) and (30), re-spectively.Remarkably, the above bound is tight in the sense thatfor all states that R ⊥ = R , the bound is achieved. Thishappens when T t ~x = ~y = 0, gives therefore a sufficientcondition for the reachable upper bound of the quantumdiscord. In this case t becomes the largest eigenvalueof T t T and ˆ e is the corresponding eigenvector. Sur-prisingly, as we will show in the example below, it mayhappens R ⊥ = R but the upper bound (27) is achieved.In such cases the absolute minimum of the conditionalentropy happens for some vectors in R ⊥ ⊆ R . Recentlyan upper bound for the quantum discord is obtained in[17] as Q ( ρ AB ) ≤ S ( ρ B ). A comparison of this with theupper bound presented in (30) shows that for all statesfor which S ( ρ AB ) − h (cid:18) √ x + t (cid:19) >
0, our bound isstronger.As an illustrative example let us consider a two-parameter class of states discussed in [22] ρ AB ( a, b ) = 12 a a − a − b − a + b a a , (34)where 0 ≤ a ≤ a − ≤ b ≤ − a . The discord ofthis state is [22] Q ( ρ AB ( a, b )) = min { a, q } , (35)where q = a (cid:20) a (1 − a ) − b (cid:21) − b (cid:20) (1 + b )(1 − a − b )(1 − b )(1 − a + b ) (cid:21) − √ a + b " √ a + b − √ a + b + 12 log (cid:20) − a ) − b )(1 − b )(1 − a − b ) (cid:21) . (36)For this state we get ~x = − ~y = − b , T = a − a
00 0 2 a − . (37)Clearly, T t ~x = − (2 a − ~y , so that for all values of a and b = 0 we have R = span { ~y } . Therefore an arbitraryelement of R ⊥ can be written as ˆ e = (cos φ, sin φ, t .In this case we get ˆ e t0 T t T ˆ e = a , which is independentof φ , so that t = a . This means that every vector inthe two-dimensional subspace R ⊥ , corresponding to the xy -plane, gives the desired upper bound. Therefore, weobtain min ˆ e ∈R ⊥ S ( ρ A |{ ˆ e } ) = h √ a + b ! , (38)and S ( ρ A ) = S ( ρ B ) = h (cid:18) b (cid:19) , (39) S ( ρ AB ) = h (cid:18) a, − a − b , − a + b (cid:19) , (40)where h ( µ , µ , µ ) = − P i =1 µ i log µ i . After some cal-culations we find that the inequality (30) leads to Q ( ρ AB ( a, b )) ≤ q, (41)where q is defined in Eq. (36). A comparison of thiswith Eq. (35) shows that for some region of parameters,namely when q ≤ a , our upper bound is tight. Figs. 1and 2 illustrate this fact. These figures also reveal thatfor this class of states our bound is stronger than theupper bound introduced in Ref. [17]. IV. OPTIMIZATION
In this section we present an analytical procedure foroptimization of the conditional entropy for a general two-qubit state. We also provide example for which one canobtain the minimum, analytically. In order to determinethe minimum of the conditional entropy (19), we have tocalculate its derivatives with respect to θ and φ . To dothis we need to calculate derivatives of the probabilitiesgiven by Eqs. (15) and (21) with respect to θ and φ . For H a L D (cid:144) upp e r bound H b L D (cid:144) upp e r bound H c L D (cid:144) upp e r bound H d L D (cid:144) upp e r bound FIG. 1. (Color online) Quantum discord [dashed-blue lines],our upper bound [solid-black lines], and the upper bound ofRef. [17] [dotted-red lines] are plotted versus a for the state ρ AB ( a, b ) with: (a) b = 0 .
1, (b) b = 0 .
3, (c) b = 0 .
5, and (d) b = 0 . H a L - D (cid:144) upp e r bound H b L - - - D (cid:144) upp e r bound H c L - - D (cid:144) upp e r bound H d L - - D (cid:144) upp e r bound FIG. 2. (Color online) Quantum discord [dashed-blue lines],our upper bound [solid-black lines], and the upper bound ofRef. [17] [dotted-red lines] are plotted versus b for the state ρ AB ( a, b ) with: (a) a = 0 .
1, (b) a = 0 .
3, (c) a = 0 . a = 0 .
9. In the cases (c) and (d), our upper bound completelycoincide with the quantum discord. instance, their derivative with respect to θ are as follows ∂p , ∂θ = ±
12 ˆ n t ,θ ~y, (42) ∂w , ∂θ = 14 ˆ n t ,θ h ~y ± T t ˆ Z + i , (43) ∂w , ∂θ = 14 ˆ n t ,θ h − ~y ± T t ˆ Z − i , (44)with ˆ Z + = T ˆ n + ~x | T ˆ n + ~x | , ˆ Z − = T ˆ n − ~x | T ˆ n − ~x | , (45)and the unit vector ˆ n ,θ is defined byˆ n ,θ = ∂ ˆ n∂θ = (cos θ cos φ, cos θ sin φ, − sin θ ) t . (46)Evidently ˆ n · ˆ n ,θ = 0. By defining the nonunit vector ˜ n ,φ by ˜ n ,φ = ∂ ˆ n∂φ = ( − sin θ sin φ, sin θ cos φ, t , (47)orthogonal to both ˆ n and ˆ n ,θ , we get a similar equationsfor the derivatives of the probabilities with respect to φ ,but now ˆ n ,θ is replaced by ˜ n ,φ . Finally using the aboveequations, we find the following relations for derivativesof the conditional entropy ∂S ( ρ A |{ Π Bk } ) ∂θ = −
14 ˆ n t ,θ ~A, (48) ∂S ( ρ A |{ Π Bk } ) ∂φ = −
14 ˜ n t ,φ ~A (49)where ~A is a vector defined by ~A = (cid:20) log w w p w w p (cid:21) ~y + (cid:20) log w w (cid:21) T t ˆ Z + + (cid:20) log w w (cid:21) T t ˆ Z − . (50)Equations (48) and (49) enable one to present the fol-lowing theorem as a necessary and sufficient conditionfor vector ˆ n = (sin θ cos φ, sin θ sin φ, cos θ ) t to be thestationary measurement of the conditional entropy, i.e. ∂S ( ρ A |{ Π Bk } ) /∂ ˆ n = 0, as follows Theorem 2.
Vector ˆ n ∈ R is a stationary measurementfor the quantum conditional entropy if and only if vector ~A , defined by Eq. (50), satisfy the following condition ˆ n t ⊥ ~A = 0 , (51) where ˆ n ⊥ is any vector perpendicular to ˆ n , i.e. ˆ n t ⊥ ˆ n = 0 . Remark 1.
Note that if we proceed the optimizationprogress by using the Lagrange multiplier λ , having ˆ n t ˆ n =1 as a constraint, we find for the stationary condition d (cid:2) S ( ρ A |{ Π Bk } ) − λ (ˆ n t ˆ n − (cid:3) = 0 the following relation ~A = A ˆ n, (52) where ~A is defined in Eq. (50), and A is given by A = − (cid:20) log w w (cid:21) ~x t ˆ Z + + (cid:20) log w w (cid:21) ~x t ˆ Z − − S ( ρ A |{ Π Bk } ) − (cid:20) log w w w w p p (cid:21) . (53)The stationary condition (52) is equivalent to thatgiven by Eq. (51) in the sense that both conditions re-quire that in the extremum points, vector ~A should bedirected to ˆ n . Unfortunately, these conditions do nothave simple solutions, for ~A as well as A depends also onˆ n . Moreover, knowing the extremum points of the con-ditional entropy is not enough to establish its minimum,and we need, in addition, to evaluate the conditional en-tropy in the extremum points to find the minimum one.Below we exemplify these conditions for one particular class of states where we have already obtained the mini-mum of the conditional entropy without optimization. States with T t ~x = 0 and ~y = 0 .— For this class of stateswe have already shown that the optimal measurement liesin the direction of the eigenvector of T t T , correspondingto the largest eigenvalue. We now reconsider this classof states and obtain the optimum measurement by usingthe optimization condition given above. For this class ofstates vector ~A takes the following form ~A = 2 | T ˆ n + ~x | (cid:20) log | T ˆ n + ~x | − | T ˆ n + ~x | (cid:21) T t T ˆ n. (54)It is clear that the condition (51) leads to ˆ n t ⊥ T t T ˆ n = 0,which has solutions when ˆ n is an eigenvector of T t T . Butthe 3 × T t T has three nonnegative eigenvalues { t , t , t } corresponding to the eigenvectors { ˆ e , ˆ e , ˆ e } .Therefore, in this particular case, three eigenvectors of T t T are stationary measurements of the conditional en-tropy. For these directions, conditional entropy takes thefollowing form S ( ρ A |{ ˆ e k } ) = h p x + t k ! , for k = 1 , , . (55)Simple evaluation shows that the minimum of the aboveequation happens when t k corresponds to the largesteigenvalue of T t T .Example above show that the stationary conditiongiven in Eq. (51) gives us, in general, more than onesolution for the measurement direction ˆ n , and we haveto find the optimal one by further evaluations. However,the presented stationary condition is simple and com-putationally straightforward, in the sense that it can bestored in a computer and that could be used for doingsymbolic and numerical calculations. V. CONCLUSION
All difficulties in calculating the quantum discord arisefrom the difficulty in finding the minimum of the quan-tum conditional entropy S ( ρ AB |{ Π Bk } ) over all measure-ments on the subsystem B . In this work, we have pre-sented a simple relation for the quantum conditional en-tropy of a two-qubit system, as the difference of twoShannon entropies. Using it, we have obtained the quan-tum discord for a class of states for which the conditions T t ~x = 0 and ~y = 0 are satisfied. This class of statesincludes the Bell-diagonal states, a three-parameter sub-class of X states, and some zero-discord states. Al-though, the quantum discord of these states is already ob-tained analytically, but the presented form for the condi-tional entropy enables one to obtain the previous resultsin a simpler manner. For these states, in particular, itis shown that the quantum conditional entropy reducesto the binary Shannon entropy, so that in minimizingthe conditional entropy we encountered with the simpleproblem of minimizing binary Shannon entropy.We have also shown that such obtained relation for theconditional entropy can be used to provide a computabletight upper bound on the quantum discord, so that thequestion of how large the quantum discord can possiblybe, can be answered more reasonably. We have presentedsufficient conditions for the reachable upper bound of thequantum discord and have exemplified this bound for atwo-parameter class of states and have shown that thebound may be tight even in the absence of sufficient con-ditions.Based on the simple form of the conditional entropy,a general procedure of optimization of the conditionalentropy is also given, and conditions under which con- ditional entropy is stationary are presented. We showthat our algorithm of optimization is efficient in the sensethat it can be used to calculate quantum discord for someclasses of states, analytically. The presented stationarycondition is simple and computationally straightforward,in the sense that it can be stored in a computer and thatcould be used for doing symbolic and numerical calcula-tions. The paper, therefore, can be regarded as a furtherdevelopment in the calculation of the quantum discordfor an arbitrary state of two-qubit system. The methodpresented in this paper can be generalized to higher di-mensional systems. [1] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. ,777 (1935)[2] E. Schr¨odinger, Naturwiss. , 807 (1935).[3] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. , 017901(2001).[4] L. Henderson, and V. Vedral, J. Phys. A , 6899 (2001).[5] W. H. Zurek, Phys. Rev. A , 012320 (2003).[6] Knill E. and Laflamme R., Phys. Rev. Lett. ,5672(1998).[7] A. Datta, A. Shaji and C.M. Caves, Phys. Rev. Lett. , 042325(2009).[8] B. P. Lanyon, M. Barbieri, M. P. Almeida, and A. G.White, Phys. Rev. Lett. , 042303 (2008).[10] M. D. Lang, and C. M. Caves, Phys. Rev. Lett. ,150501 (2010).[11] M. Ali, A. R. P. Rau, and G. Alber, Phys. Rev. A ,042105 (2010).[12] Q. Chen, C. Zhang, S. Yu, X. X. Yi, and C. H. Oh, Phys.Rev. A , 042313 (2011).[13] M. Shi, W. Yang, F. Jiang, and J. Du, J. Phys. A: Math. Theor. , 415304 (2011).[14] L. X. Cen, X. Q. Li, J. Shao, and Y. J. Yan, Phys. Rev.A , 054101 (2011).[15] G. Adesso and A. Datta, Phys. Rev. Lett. , 030501(2010).[16] D. Girolami, and G. Adesso, Phys. Rev. A , 052108(2011).[17] Z. Xi, X-M Lu, X. Wang, and Y. Li, J. Phys. A: Math.Theor. , 375301 (2011).[18] S. Yu, C. Zhang, Q. Chen, C. H. Oh, arXiv:1102.1301(2011).[19] R. Horodecki, and M. Horodecki, Phys. Rev. A , 1838(1996).[20] R. Gilmore, Lie Groups, Lie Algebras, and Some of TheirApplications (John Wiley & Sons, New York, USA, 1974).[21] M. A. Nielsen, and I. L. Chuang,
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