aa r X i v : . [ qu a n t - ph ] N ov Noname manuscript No. (will be inserted by the editor)
Phase diagram for the one-way quantum deficit oftwo-qubit X states
M.A.Yurischev
Received:
Abstract
The one-way quantum deficit, a measure of quantum correlation,can exhibit for X quantum states the regions (subdomains) with the phases ∆ and ∆ π/ which are characterized by constant (i.e., universal) optimal mea-surement angles, correspondingly, zero and π/ z -axis anda third phase ∆ ϑ with the variable (state-dependent) optimal measurementangle ϑ . We build the complete phase diagram of one-way quantum deficit forthe XXZ subclass of symmetric X states. In contrast to the quantum discordwhere the region for the phase with variable optimal measurement angle isvery tiny (more exactly, it is a very thin layer), the similar region ∆ ϑ is largeand achieves the sizes comparable to those of regions ∆ and ∆ π/ . This in-stils hope to detect the mysterious fraction of quantum correlation with thevariable optimal measurement angle experimentally. Keywords
X density matrix · One-way deficit function · Domain ofdefinition · Piecewise-defined function · Subdomains · Critical lines andsurfaces
Quantum correlations play the key role in quantum information science. Manykinds of quantum correlations have been introduced so far and now their prop-erties are scrupulously analyzed. One of the most important places among suchcorrelations beyond quantum entanglement belongs to the quantum discordand one-way quantum work (information) deficit [1,2,3,4,5,6,7,8,9]. In thepresent paper we focus on the latter measure of quantum correlation. Theone-way deficit has operational interpretation in thermodynamics and is equal
M. A. YurischevInstitute of Problems of Chemical Physics, Russian Academy of Sciences, Chernogolovka142432, Moscow Region, RussiaE-mail: [email protected] M.A.Yurischev to a slightly different version of quantum discord [9] called in Ref. [1] as the“thermal discord”; this term is also supported in the topical review [3].Remarkably, both varieties of discord definition yield the same results forthe Bell-diagonal states and even for the X quantum states if one qubit of thesystem is maximally mixed and the measurements are performed on this qubit[10]. However, in spite of closeness of definitions, the quantum discord andone-way quantum deficit may lead, generally speaking, to the quite differentquantitative and actually qualitative behavior of quantum correlation in moregeneral cases.It is known [11,12,13,14] that the optimization of quantum discord as wellas one-way deficit for X states is reduced to a minimization only on one vari-able, namely the polar angle θ ∈ [0 , π/
2] (the angle of deviation from the z -axis of X state) whereas the azimuthal angle φ can be always eliminated bylocal rotations around the z -axis. A minimization procedure for measurement-dependent discord, Q ( θ ), and one-way deficit, ∆ ( θ ), leads to the optimizedfunctions of discord Q = min { Q , Q π/ , Q θ ∗ } (subscripts 0, π/
2, and θ ∗ de-noting the corresponding optimal measured angles for the quantum discord)and one-way deficit ∆ = min { ∆ , ∆ π/ , ∆ ϑ } in their domain of definitionare the piecewise-defined ones. In other words, the total domain of definitionconsists of subdomains each one corresponding to the own branch (phase orfraction - in physical language). Important problem is to separate all possiblephases of quantum correlation and find in fact the exhausted phase diagram.Recently [15] the quantum discord for the symmetric XXZ subclass of Xstates has been considered and the three-dimensional phase diagram for it hasbeen obtained. In this paper we perform calculations for the same subclassbut this time for the one-way quantum deficit. This allows us to comparethe behavior of both measures of quantum correlation and reveal differencesbetween them. One of the most surprising results is that the “anomalous”subdomain with variable optimal measurement angle is essentially larger thanthat for the quantum discord and its sizes achieve the values which make itaccessible for experimental investigations.The reminder of this paper is organized as follows. In the next section wediscuss the possibility to create the complete phase diagram for the generaltwo-qubit X quantum state. The domain of definition of one-way deficit func-tion for the symmetric XXZ quantum state is obtained in Sec. 3. All necessaryformulas for the branches of piecewise one-way deficit function are presentedin Sec. 4. Equations for the boundaries between different fractions of one-waydeficit are given in Sec. 5. In Sec. 6 we solve the equations for the boundariesand discuss the results obtained. Concluding remarks and possible perspectivesare summarized in Sec. 7. hase diagram for the one-way quantum deficit of two-qubit X states 3 In the most general case, the X quantum state of two-qubit system AB canbe written as ρ AB = 4 − (1 + s σ z ⊗ s ⊗ σ z + c σ x ⊗ σ x + c σ y ⊗ σ y + c σ x ⊗ σ y + c σ y ⊗ σ x + c σ z ⊗ σ z ) , (1)where σ α ( α = x, y, z ) is the vector of the Pauli matrices. The density matrix(1) contains seven real parameters s , s , c , c , c , c , and c which arethe unary and binary correlation functions, i.e., experimentally measurablequantities: s = h σ z i = Tr( ρ AB σ z ⊗ , s = h σ z i = Tr( ρ AB ⊗ σ z ) ,c = h σ x σ x i = Tr( ρ AB σ x ⊗ σ x ) , c = h σ y σ y i = Tr( ρ AB σ y ⊗ σ y ) ,c = h σ x σ y i = Tr( ρ AB σ x ⊗ σ y ) , c = h σ y σ x i = Tr( ρ AB σ y ⊗ σ x ) , (2) c = h σ z σ z i = Tr( ρ AB σ z ⊗ σ z ) . It is clear that − ≤ s , s , c , c , c , c , c ≤ , (3)hence in any case the parameters do not come out of the seven-dimensionalcube in the space R .The density matrix ρ AB in open form reads ρ AB = 14 s + s + c c − c − i ( c + c )0 1 + s − s − c c + c i ( c − c )0 c + c − s + s − c − i ( c − c ) c − c − s − s + c + i ( c + c ) (4)that demonstrates the obvious X structure.Quantum correlations are invariant under the local unitary transformationsof density matrix [16]. Thanks to this fundamental property one can eliminatethe complex phases in the off-diagonal elements of density matrix (4) andreduce it with the help of local rotations around the z -axis, exp( − iϕ σ z / ⊗ exp( − iϕ σ z / real X form (see [1] and, e.g., [17,18]): ρ AB = 14 s + s + c u s − s − c v v − s + s − c u − s − s + c (5) M.A.Yurischev with u = [( c − c ) + ( c + c ) ] / , v = [( c + c ) + ( c − c ) ] / . (6)This matrix contains five real parameters: s , s , c , u , and v . All kinds ofquantum correlations in the state (5) are the same as in the original seven-parameter state (1). Moreover, since off-diagonal entries are now non-negative,this will allow us to set by optimization the azimuthal angle φ = 0 [11,13]. (Infact, this nonnegativity is enough only for the product of off-diagonal entries, uv ≥ c ) − ( s + s ) ≥ ( c − c ) + ( c + c ) = u , (1 − c ) − ( s − s ) ≥ ( c + c ) + ( c − c ) = v . (7)A body which is bounded by these two intersecting surfaces of second order isthe domain of definition for the arguments of quantum correlation functions.Note that this domain lies in the seven-dimensional hypercube [ − , × .Classification and construction of all possible phases of quantum correlationin the whole seven-parameters space is an enormous task which as yet onlywaits for its solution. One way to solve it is to build an atlas of maps [19], i.e.,the collection of two- or three-dimensional phase diagrams. We now pass onto the first, likely simplest but important “map” of such an atlas. As for the quantum discord [15], in this paper we restrict ourselves by thesame three-dimensional space, i.e., set s = s (the mirror symmetry C s oftwo-qubit system), c = c = 0, and c = c [the axial spin symmetry U (1)].In this case the density matrix is written as ρ AB = 4 − [1 + s ( σ z ⊗ ⊗ σ z ) + c ( σ x ⊗ σ x + σ y ⊗ σ y ) + c σ z ⊗ σ z ] . (8)We will call this state as the symmetric XXZ one — by analogy with thewell-known statistical-mechanical Hamiltonians [20,21]. Notice that the z -component of total spin, S z = ( σ z ⊗ ⊗ σ z ) /
2, commutes with the densitymatrix (8); this is a sequence of inner symmetry U (1) = { exp( iϕS z ) | ϕ ∈ [0 , π ] } . As a result, the full symmetry group is C s × U (1).In the open form the density matrix (8) is given by ρ AB = 14 s + c − c c
00 2 c − c
00 0 0 1 − s + c . (9) hase diagram for the one-way quantum deficit of two-qubit X states 5 Fig. 1
Shaded tetrahedron T embedded in a three-dimensional cube (dotted lines) is thedomain of definition for the parameters (arguments) s , c , and c Three-parameter quantum states with such a block-diagonal structure are dis-cussed in connection with the problem of maximally entangled mixed states(MEMS) [22] (see also [23]).Restrictions (7) are reduced here to the conditions c ∈ [ − , , s ∈ [ − (1 + c ) / , (1 + c ) / , c ∈ [ − (1 − c ) / , (1 − c ) / T (see Fig. 1). This tetrahedron is enclosed inthe three-dimensional cube [ − , ⊗ , has the vertices v , v , v , and v andisosceles triangle faces. Volume of tetrahedron T equals one sixth part of cubevolume (= 2 ). So, one may say that the one-way deficit ∆( s , c , c ) is afunction on the tetrahedron T .Notice that the symmetric XXZ states (8) may be written in an equivalentform which is important for the MEMS problem [24,25], ρ AB = q | Ψ + ih Ψ + | + q | Ψ − ih Ψ − | + q | ih | + q | ih | , (11)where q + q + q + q = 1, q , = (1 ± c − c ) / , q , = (1 ± s + c ) / , (12) | Ψ ± i = ( | i ± | i ) / √ | i and | i are product-states orthogonal to | Ψ ± i ; they represent two “up” and “down” oriented spins(or two horizontally and vertically polarized photons), respectively. One mayexchange | Ψ + i ↔ | Ψ − i and | i ↔ | i because they belong to the localunitary transformations and therefore do not change the value of quantumcorrelation. Quantities q i ( i = 1 , · · · ,
4) are equal to the eigenvalues of densitymatrix and therefore must be non-negative. Hence the domain of definition
M.A.Yurischev in the space ( q , q , q ) is a three-dimensional corner restricted by four planesand conditions: q ≥ q ≥ q ≥
0, and q + q + q ≤
1. Further, arepresentation of Eq. (11) in Pauli matrices takes the form ρ AB = 14 { q + q + 2 q − σ z ⊗ ⊗ σ z )+( q − q )( σ x ⊗ σ x + σ y ⊗ σ y ) + [1 − q + q )] σ z ⊗ σ z } . (13)By the local unitary transformations, the state (11) is reduced to [26,27] ρ AB = q ′ | Φ + ih Φ + | + q ′ | Φ − ih Φ − | + q ′ | ih | + q ′ | ih | , (14)where | Φ ± i = ( | i ± | i ) / √ ρ AB = ǫ | Φ + ih Φ + | + (1 − ǫ )( m | ih | + (1 − m ) | ih | ) (15)( ǫ and m play a role of concentrations). One-way quantum deficit for a bipartite state ρ AB is defined as the minimalincrease of entropy after a von Neumann measurement on one party (withoutloss of generality, say, B ) [8] (see also, e.g., [1,5])∆ = min { Π k } S (˜ ρ AB ) − S ( ρ AB ) , (16)where S ( · ) means the von Neumann entropy and˜ ρ AB ≡ X k p k ρ kAB = X k ( I ⊗ Π k ) ρ AB ( I ⊗ Π k ) + (17)is the weighted average of post-measured states ρ kAB = 1 p k ( I ⊗ Π k ) ρ AB ( I ⊗ Π k ) + (18)with the probabilities p k = Tr( I ⊗ Π k ) ρ AB ( I ⊗ Π k ) + . (19)In Eq. (17), Π k ( k = 0 ,
1) are the general orthogonal projectorsΠ k = V π k V + , (20) hase diagram for the one-way quantum deficit of two-qubit X states 7 where π k = | k ih k | and transformations { V } belong to the special unitary group SU (2). Rotations V may be parametrized by two angles θ and φ (polar andazimuthal, respectively): V = (cid:18) cos( θ/ − e − iφ sin( θ/ e iφ sin( θ/
2) cos( θ/ (cid:19) (21)with 0 ≤ θ ≤ π and 0 ≤ φ < π .Eigenvalues of matrix (9) are equal to λ , = (1 ± s + c ) / , λ , = (1 ± c − c ) / . (22)Using these equations one gets the pre-measurement entropy function S ( s , c , c ) = h ((1+2 s + c ) / , (1 − s + c ) / , (1+2 c − c ) / , (1 − c − c ) / , (23)where h ( x , x , x , x ) = − x log x − x log x − x log x − x log x withcondition x + x + x + x = 1 is the quaternary entropy function. In explicitform, S ( s , c , c ) = 2 ln 2 −
14 [(1 + 2 c − c ) ln(1 + 2 c − c ) + (1 − c − c ) ln(1 − c − c )+(1 + 2 s + c ) ln(1 + 2 s + c ) + (1 − s + c ) ln(1 − s + c )] . (24)The quantum state (9) after measurements Π k ( k = 0 ,
1) and averaging inaccordance with Eq. (17) takes the form˜ ρ AB = 14 s + ( s • • • + c ) cos θ ( s + c ) e iφ sin2 θ − c + ( s • • + c ) sin θ c e iφ sin2 θ c sin θ − c − ( s •− c ) sin θc e iφ sin θ − c e iφ sin2 θ ( s − c ) e iφ sin2 θ − s − ( s − c ) cos θ (25)(for the sake of simplicity, the bold points are put instead of correspondingcomplex conjugated matrix elements of the Hermitian matrix ˜ ρ AB ).We used the Mathematica software to extract the eigenvalues of matrix(25). First of all we established that the secular equation for the given matrix isfactorized into product of two polynomials of second orders. In proving this keyproperty, the following trick turned out useful. Namely, we first transformedthe matrix elements to the exponential form (the command TrigToExp), foundthe secular equation, factorized it, then returned by applying the command M.A.Yurischev
ExpToTrig to the trigonometric expressions and finally simplified the resultusing the command FullSimplify. As a result, we arrived at the eigenvalues ofpost-measurement state (25), Λ , = 14 [[1 + s cos θ ± { ( s + c cos θ ) + c sin θ } / ]] , (26) Λ , = 14 [[1 − s cos θ ± { ( s − c cos θ ) + c sin θ } / ]] . It is seen that the azimuthal angle φ has dropped out from the given expres-sions. This is a general property and occurs every time when only one of theoff-diagonals of the density matrix is non-zero [12,30]. Hence, the optimizationreduces to that in a single variable θ .Using Eqs. (26) we find the post-measured entropy (entropy after measure-ment) ˜ S ( θ ) ≡ S (˜ ρ AB ) = h ( Λ , Λ , Λ , Λ ) , (27)where h ( · ) is again the quaternary entropy function. The function ˜ S of ar-gument θ is invariant under the transformation θ → π − θ therefore it isenough to restrict oneself by values of θ ∈ [0 , π/ S and ˜ S , as functions of s and c , are symmetricunder the reflections s → − s and c → − c .Notice the following. The post-measument entropy ˜ S is related to the con-ditional entropy S cond [see Eqs. (25)-(26) in Ref. [15]] by equation (see [31]and also Appendix in Ref. [15])˜ S ( θ ) = S cond ( θ ) + h ((1 + s cos θ ) / , (28)with h ( x ) = − x ln x − (1 − x ) ln(1 − x ) being Shennon’s binary entropy function.The first derivative of h -term with respect to θ equals zero at both endpoints θ = 0 and π/
2. Hence if the first derivative of S cond ( θ ) with respect to θ vanishes at θ = 0 and π/ S ( θ ) and vice versa. Moreover, in accordance with Eq. (28), the non-optimized one-way deficit can be written as ∆ ( θ ) = h ((1 + s cos θ ) / − S ( ρ AB ) + S cond ( θ ) . (29)On the other hand, the measurement-dependent quantum discord can be pre-sented as Q ( θ ) = h ((1 + s cos θ ) / | θ =0 − S ( ρ AB ) + S cond ( θ ) (30)because h ((1 + s cos θ ) / | θ =0 = S ( ρ B ), where ρ B = Tr A ρ AB . So, Eqs. (29)and (30) show a very close mathematical relation between both measures ofquantum correlation: the only difference is either we take h -term with arbitrary θ or set θ = 0. This relation is valid for general X states. hase diagram for the one-way quantum deficit of two-qubit X states 9 From Eqs. (26) and (27), one gets an expression for the post-measuremententropy:˜ S ( θ ; s , c , c ) = 2 ln 2 − X m,n =1 (cid:18) − m s cos θ + ( − n q ( s + ( − m c cos θ ) + c sin θ (cid:19) × ln (cid:18) − m s cos θ + ( − n q ( s + ( − m c cos θ ) + c sin θ (cid:19) . (31)In the following analysis, this function will serve us to probe and identify thetypes of subregions in phase diagrams. The function ˜ S ( θ ; s , c , c ) is differen-tiable at any point θ and, in full conferment with the statement made in theIntroduction, its first derivatives with respect to θ identically equal zero for ∀ s , c , c ∈ T at both ends of the interval [0 , π/ θ = 0 is given as˜ S ≡ ˜ S (0) = 2 ln 2 −
12 (1 − c ) ln(1 − c ) −
14 [(1 + 2 s + c ) ln(1 + 2 s + c ) + (1 − s + c ) ln(1 − s + c )] (32)and at the endpoint θ = π/ S π/ ≡ ˜ S ( π/
2) = 2 ln 2 − (cid:20)(cid:18) q s + c (cid:19) ln (cid:18) q s + c (cid:19) + (cid:18) − q s + c (cid:19) ln (cid:18) − q s + c (cid:19)(cid:21) . (33)Equations (23),(26), and (27) define the measurement-dependent one-waydeficit function ∆ ( θ ) = ˜ S ( θ ) − S. (34)In the wake of ˜ S ( θ ), the first derivative of ∆ ( θ ) with respect to θ is identicallyequal to zero at both endpoints θ = 0 and θ = π/ S ′ (0) = ∆ ′ (0) ≡ , ˜ S ′ ( π/
2) = ∆ ′ ( π/ ≡ . (35)By analogy with the quantum discord [17,18,30,32] (see also [15]), theone-way quantum deficit may be written as almost closed analytical formula[33] ∆ = min { ∆ , ∆ π/ , ∆ ϑ } . (36)Thus, it can consist of three branches. However, it should be noted that ingeneral these branches may further out split into new subbranches.Using the above equations one obtains the expression for the 0-branch: ∆ ( s , c , c ) = −
12 (1 − c ) ln(1 − c )+ 14 [(1 + 2 c − c ) ln(1 + 2 c − c ) + (1 − c − c ) ln(1 − c − c )] . (37) Surprisingly, this branch is identical to the similar branch of discord Q [15],i.e., ∆ = Q . Moreover the function ∆ is symmetric under the reflection c → − c and does not depend on s .For the branch ∆ π/ with the π/ ∆ π/ ( s , c , c ) = − (cid:20)(cid:18) q s + c (cid:19) ln (cid:18) q s + c (cid:19) + (cid:18) − q s + c (cid:19) ln (cid:18) − q s + c (cid:19)(cid:21) + 14 [(1 + 2 c − c ) ln(1 + 2 c − c ) + (1 − c − c ) ln(1 − c − c )+(1 + 2 s + c ) ln(1 + 2 s + c ) + (1 − s + c ) ln(1 − s + c )] . (38)This function is symmetric both on s and c .The third and last branch ∆ ϑ is obtained by numerically solving the one-dimensional optimization problem for the function ˜ S ( θ ).We turn now to the discussion of boundaries distinguishing the phases ofquantum correlation. The problem of phases and boundaries between them is well known in ther-modynamics and statistical physics [20,34] as well as in other fields of science.Classification of phases depends on characteristic features which are put in itsground. Therefore it is not surprising that, e.g., one discovers more and morenew phases for water.Quantum correlations are piecewise-defined functions. A choice of a cer-tain branch in the given point of parameter space is dictated by a minimumcondition like (36). But how to decompose the whole domain of definition ofquantum correlation function into subdomains? The answer to this questionwas firstly given in Refs. [17,18] (see also [30,32]) with the quantum discord.Applying those ideas to the one-way quantum deficit, one may say thefollowing. On the one hand, the boundary between the phases with zero and π/ S = ˜ S π/ or ∆ = ∆ π/ . (39)When crossing this boundary, the optimal measurement angle experiences thejump ∆ϑ = π/ ϑ -phase to the 0- or π/ θ = 0 and π/
2. In these cases the equations for theboundaries are reduced to a requirement of vanishing the second derivativesof post-measurement entropy or measurement-dependent one-way deficit withrespect to θ at corresponding endpoints [14,33]˜ S ′′ (0) = 0 or ∆ ′′ (0) = 0 (40) hase diagram for the one-way quantum deficit of two-qubit X states 11 for the 0-boundary and˜ S ′′ ( π/
2) = 0 or ∆ ′′ ( π/
2) = 0 (41)for the π/ ∆ϑ = 0.For the quantum states under discussion the second derivatives at end-points follow from Eq. (31) and equal˜ S ′′ (0) = 14 (cid:26) s ln 1 + 2 s + c − s + c + c ln (1 + 2 s + c )(1 − s + c )(1 − c ) − c (cid:20) s + c ln 1 + 2 s + c − c + 1 s − c ln 1 − c − s + c (cid:21)(cid:27) (42)and˜ S ′′ ( π/
2) = c s + c − c r ln 1 + r − r − s (cid:18) (1 + c /r ) r + (1 − c /r ) − r (cid:19) (43)with r = p s + c .Equations (39), (40), and (41) together with expressions (32), (33), (37),(38), (42), and (43) are transcendental and they can be solved numerically bythe bisection method.The above mechanism of arising the boundaries between phases is con-firmed for the cases when the post-measurement entropy function is unimodal.However, most recently [33] a bimodal behavior for the post-measurement en-tropy function has been found and a new important equation has been addedto the collection of boundary equations, namely˜ S = ˜ S ϑ or ∆ = ∆ ϑ . (44)These conditions reflect a jump (sudden change) of optimal measurement anglefrom zero to a finite step ∆ϑ > ∆ϑ = π/
2. For definiteness, we will callsuch a boundary as the 0 ′ -one.Generally speaking the jumps between the endpoint θ = π/ ′ -boundaryalso by the bisection method with searching the interior minimum of post-measurement entropy or measurement-dependent one-way quantum deficit atevery step of iteration procedure by the golden section method.The above classification of phases is based on the type of optimal mea-surement angle. But in general such an approach does not exhaust all possiblebranches of piecewise-defined function. New subbranches can appear from thesplitting of some original branches. For instance, such a phenomenon was pre-viously observed for the π/ Q π/ , which decays into twosubbranches Q (1) π/ and Q (2) π/ in the limit of Bell-diagonal states (see Fig. 7b inRef. [32] and Sec. 2.1 in Ref. [15]). The reason is simple: when s = s = 0, the function Q π/ ( c , c , c ) by | c | 6 = | c | contains inside itself the piecewise func-tions like | c − c | . In similar situations, the additional boundaries are revealedfrom the detection of fracture points, i.e., from the condition of discontinuityfor the first derivatives with respect to the parameters of the model:∆ ′ x ( x − = ∆ ′ x ( x + 0) , x ∈ { s , c , c } . (45)However, we did not observe such cases in the present paper because c = c . The task now is to separate the body T by using boundary equations into sub-domains each one corresponding to a certain branch according to the minimumcondition (36). Our strategy is reduced to the following. We will take differenttwo-dimensional sections of tetrahedron T , solve the equations for boundaries,and then visually analyze the shapes of curves ˜ S ( θ ) or ∆ ( θ ). The position ofglobal minimum of these curves will give us the answer to the question aboutthe type of branch in the given part of section.The one-way deficit function ∆( s , c , c ) is invariant under the transfor-mations s → − s and c → − c . Hence the body T with all subdomains issymmetric relative to the mirror reflections in the planes s = 0 and c = 0.Owing to this, it is enough to study the phases only in a quarter of tetrahedron T .6.1 Phase diagrams on facesWe begin the analysis with the faces of tetrahedron T . Due to the mirrorsymmetry, it is enough to study the phases only on two adjacent semi-faces.Without loss of generality, we consider the faces v v v and v v v (see Fig. 1).Both halves of these two adjacent faces are shown in Fig. 2 in an unfolded form.Consider first the upper triangle. Here c = − (1 − c ) /
2. Solution of equa-tions from the previous section leads to the curves labeled in Fig. 2 by symbols0 and 1 which correspond to the 0- and π/ S ( θ ) allows to identifythe types of separate subdomains; they are marked in Fig. 2 by correspondingnames of branches. Three found types of post-measurement entropy shapes(monotonically decreasing, unimodal, and monotonically increasing) are illus-trated in Fig. 3. More complicated shapes of ˜ S ( θ ) are absent on this face.Phase diagram on the lower triangle part of Fig. 2 [here s = (1+ c ) /
2] cor-responds to the case which has been considered in Ref. [33] (Fig. 7 there); theonly difference is the coordinates ( c , c ) instead ( q , q ). Here the bimodal be-havior of ˜ S ( θ ) takes place and, correspondingly, the 0 ′ -boundary on which theoptimal measurement angle discontinuously changes its value exists betweenthe points “a” and “c”. The curve 0 ′ is a line of continuously varying ∆ϑ in hase diagram for the one-way quantum deficit of two-qubit X states 13 -1-0.500.51-1 -0.8 -0.6 -0.4 -0.2 0 c c s v v ∆ ∆ ∆ π/ ∆ ϑ abc ′ s ssss Fig. 2
Phase diagram of one-way quantum deficit on the faces of tetrahedron T . Thediagonal dotted line v v dividing the figure area in two triangle parts is the correspondingedge of the tetrahedron. The upper triangle is a half of the face v v v while the lowerone is a half of the face v v v . (The latter represents the phase diagram [33] in newvariables.) The points “a” and “b” lie on the diagonal v v , their coordinates ( c , c ) beingequal ( − . ,
0) and ( − . , − . − . , − . the limits from 0 to π/ ∆ = ∆ π/ .It is interesting to compare the location of phases ∆ , ∆ π/ , and ∆ ϑ in thetetrahedron T with that of similar discord phases Q , Q π/ , and Q θ ∗ foundearlier for the same quantum states [15]. One can see the phases of one-waydeficit and discord on faces of T in Fig. 4. Because the one-way quantumdeficit must be identical to the quantum discord for the Bell-diagonal states,we satisfy ourselves that really ∆ = Q in the case s = 0. More and above, weobserve that, due to the equality ∆ = Q , the one-way deficit ∆ and discord Q are equal when the ∆ and Q regions coincide. This circumstance considerably ˜ S, bit θ π/ a ˜ S, bit θ π/ b ˜ S, bit θ π/ c ˜ S, bit θ π/ d Fig. 3
Evolution of post-measurement entropy shapes by pass along the line c = 0 . v v v [i.e, when c = (1 − c ) / s = 0 . enlarges the common part of total domain T , where both measures of quantumcorrelation yield the same result.6.2 Phase diagrams inside the tetrahedron T Consider now the phase structure in the interior of tetrahedron. The studywill be performed by scanning the body T by taking cross-sections with planes c = const . We will go from the top to the bottom of the tetrahedron.When c = 1, i.e., on the edge v v , all quantum correlations vanish becausehere c = 0 and the system is purely classical.In the subsequent investigation it will be convenient to consider five sepa-rate intervals for c taking into account the variation of phases in longitudinaldirection (see Fig. 2).First of all the calculations show that the one-way deficit equals ∆ = ∆ in the band 1 / ≤ c ≤ hase diagram for the one-way quantum deficit of two-qubit X states 15 Fig. 4 (Color online) An outward appearance of tetrahedron T for the one-way deficit (left,see also Fig. 2) and discord (right). The regions ∆ and Q are shown by the blue colorwhile the ∆ π/ and Q π/ ones are shown by the green color. The region ∆ ϑ is yellow-coloredand because the region Q θ ∗ lies inside the tetrahedron it is not seen -1-0.500.51-1 -0.5 0 0.5 1 c s a ∆ ∆ π/ ∆ π/ -1-0.500.51-1 -0.5 0 0.5 1 c s b Q Q Q Q π Q π Fig. 5
Phase diagrams of one-way deficit (a) and discord (b) by c = 0 .
1. Subregions ∆ ϑ lie between the pairs of solid lines in a graph (a). Similar subregions for the quantumdiscord, Q θ ∗ , are thin too and therefore their locations are shown only schematically bydouble solid-dotted lines in a graph (b) (see also Fig. 3 in [15]) By c < /
3, the other two phases appear in the slices. Typical phasediagram for the one-way deficit in the interval of c from 1/3 to 0 is shown inFig. 5a for the cross-section c = 0 .
1. Phase diagram for the discord is shownin Fig. 5b for a comparison. One sees the significant differences between bothmeasures of quantum correlation. However they are the same on the line s = 0(Bell’s case) and in common parts of regions ∆ and Q .Because the phase diagrams in cross-sections are symmetric about the s and c axes, one may only focus on a quarter of the diagram. Moreover, ascalculations show, the boundaries between phases of one-way deficit lie in c s c = 0 . π/ ∆ ∆ ϑ c s c = 0 01∆ π/ ∆ ∆ ϑ Fig. 6
Fragments of phase diagrams in the sections c = 0 . c = 0 (onthe right). The curves 0 represent the 0-boundaries while the lines 1 are the π/ the regions c ≥ | c | therefore it is enough to restrict oneself by the strips | c | ≤ c ≤ (1 − c ) /
2. Left part of Fig. 6 shows the subregion by c = 0 . s = 0 . ∆ ϑ segment equals 0.008639 in the absolute units. Hence the relativearea of the whole ∆ ϑ region to the area of cross-section rectangle is 3.5%.The similar area for the discord Q θ ∗ equals 5.5 × − % only. So, one mayascertain that the region with state-dependent optimal measurement angle isexperimentally accessible for the one-way deficit but not for the discord.Phase diagram in the cross-section c = 0 is also depicted in Fig. 6. Theboundary 0 has reached here the right upper corner of the cross-section rect-angle, i.e., the vertex with coordinates (0 . , . c = s . The endpointof π/ s = 0 . ∆ ϑ achieves now 4.2%.The third interval of c ranges from c = 0 to c = − . π/ ′ -boundary, i.e., instead of the appearance of interior minimum via bifurcation,there is now observed a bimodal behavior of curve ˜ S ( θ ) (see Fig. 8) that isaccompanied with the discontinuous change of optimal measurement angle onthe critical line 0 ′ . Figure 8a fixes the moment when the interior minimumachieves the value of post-measurement entropy at the angle θ = 0. Betweenthe lines 0 and 0 ′ the interior minimum is lower than the minimum at θ = 0as shown in Fig. 8b; hence the fraction ∆ ϑ exists here.The next interval − . < c < − . hase diagram for the one-way quantum deficit of two-qubit X states 17 c s c = − . π/ ∆ a ∆ ϑ ← c s c = − . ϑ ∆ ′ ← b Fig. 7
Phase diagram by c = − . ′ -boundary. The arrows on the right vertical sides ( s = 0 .
4) mark thepoint c = 0 . ′ -boundary ˜ S, bit θ a ˜ S, bit θ b Fig. 8
Post-measurement entropy ˜ S vs θ by c = − . c = 0 . s = 0 . by c = − . c reaches the value of − . ′ meet each other on the side s = (1 + c ) / c , the π/
2- and 0 ′ -lines intersect insidethe body T and the boundary ∆ = ∆ π/ appears (see Fig. 10). The post-measured entropy curve near the critical line 2 has the interior maximum .In the limit c → −
1, i.e., on the edge v v of tetrahedron T , the valueof parameter s vanishes, the region ∆ ϑ disappears, and the quantum statetransforms into the Bell-diagonal one. On this edge the optimized one-waydeficit and discord coincide and are given by∆ = Q = 12 [(1 + c ) ln(1 + c ) + (1 − c ) ln(1 − c )] . (46)In the middle of the edge, c = 0, the quantum correlation is zero while at thevertecies v and v ( c = ±
1) it is, vice versa, maximal and equals 1 bit. c s c = − . π/ ∆ ← a c s c = − . π/ ∆ ϑ ∆ ′ b ← Fig. 9
Phase diagram in the cross-section c = − . c = 0 . ′ -boundary on theline s = 0 . c s a c = − . π/ ∆ ′ s c s b c = − . π/ ∆ ′ s Fig. 10
Phase diagram by c = − . ′ correspondrespectively to the π/
2- and 0 ′ -boundaries while the line 2 is the ∆ = ∆ π/ boundary. Thefraction ∆ ϑ lies between the critical lines 1 and 0 ′ . Black circle is a triple point In this paper we have investigated the three-dimensional phase diagram ofone-way quantum deficit for the XXZ family of symmetric X quantum states.The set of Figs. 2, 4, 5, 6, 7, 9, and 10 provides insight into a complete pictureof all typical features in the phase diagram. It is probable that an applicationof technologies like holography or virtual reality would be suitable for thevisualization of such 3D diagrams.It has been established that there are three branches of one-way deficitfunction and hence three different subdomains corresponding them. Two ofwhich, ∆ and ∆ π/ , are characterized by constant measurement angles —zero and π/
2, respectively. At the same time, the third region ∆ ϑ is charac- hase diagram for the one-way quantum deficit of two-qubit X states 19 terized by non-universal behavior of optimal measurement angle ϑ because itcontinuously varies with the parameters of density matrix.We have found that three possible regions of one-way deficit can be sepa-rated by four kinds of boundaries: 0 or 0 ′ which divide the ∆ and ∆ ϑ frac-tions, π/ ∆ π/ and ∆ ϑ phases, and lastly the ∆ = ∆ π/ boundary between regions ∆ and ∆ π/ . The optimal measurement angle ϑ iscontinuous when crossing the 0- and π/ ∆ϑ = π/ ∆ = ∆ π/ boundary and varies inthe limits from zero to π/ ′ .A comparison of behavior of the one-way quantum deficit and discordshows quantitative and qualitative difference between those measures of quan-tum correlation in general. However they are identical for the Bell-diagonalstates, i.e., in the plane s = 0. Moreover, we have established that both mea-sures of quantum correlation coincide in the intersection of regions ∆ and Q ( ∆ ∩ Q ). (This result is valid for general X states.) But quite a difference inother cases allows to say that the named quantities represent different kinds ofquantum correlation. This situation is similar to the one taking place for themean value of numbers and its various types: the arithmetic mean, harmonicmean, and so on. By this, each kind of mean has its own application.We have discovered that the region with variable optimal measurementangle is in several orders larger for the one-way quantum deficit in comparisonwith the quantum discord.It should be noted that all quantum correlations are certain functions ofordinary statistical correlations (2) [see, e.g., Eqs. (37) or (38)]. This is asequence of the fact that any quantum correlation is defined by the systemdensity matrix but its entries are expressed through the statistical correlationfunctions.We have restricted ourselves only by one phase diagram of full atlas. Thework started here should be continued to cover by separate diagrams the totalseven-dimensional space of X-state parameters. Acknowledgment
I am grateful to Dr. A. I. Zenchuk for his valuable remarks.
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