Analytic Expression of Geometric Discord in Arbitrary Mixture of any Two Bi-qubit Product Pure States

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1 Commun. Theor. Phys. 63 (2015) Vol. 63 No. 4 April Analytic Expression of Geometric Discord in Arbitrary Mixture of any Two Bi-qubit Product Pure States XIE Chuan-Mei ( Ö) 1 LIU Yi-Min ( ) 2 XING Hang (¼ ) 1 and ZHANG Zhan-Jun ( ) 1 1 School of Physics & Material Science Anhui University Hefei China 2 Department of Physics Shaoguan University Shaoguan China (Received October ; revised manuscript received December ) Abstract Quantum correlations in a family of states comprising any mixture of a pair of arbitrary bi-qubit product pure states are studied by employing geometric discord [Phys. Rev. Lett. 105 (2010) ] as the quantifier. First the inherent symmetry in the family of states about local unitary transformations is revealed. Then the analytic expression of geometric discords in the states is worked out. Some concrete discussions and analyses on the captured geometric discords are made so that their distinct features are exposed. It is found that the more averagely the two bi-qubit product states are mixed the bigger geometric discord the mixed state owns. Moreover the monotonic relationships of geometric discord with different parameters are revealed. PACS numbers: Mn Ud Key words: mixture of a pair of arbitrary bi-qubit product pure state geometric discord inherent symmetry of local unitary transformation mixing weight features 1 Introduction In 2001 Ollivier and Zurek [1] found an intriguing phenomenon that quantum correlations occur in some separable states where no quantum entanglements exist at all. They termed such quantum correlation different from entanglement as quantum discord. Quantum discord is quite striking but actually a little puzzling. How to recognize it should have attracted much attention at that time. A few years later some other works concerning quantum correlation different from entanglement also appeared gradually. In particular the physics recognitions and potential applications [2 16] of quantum correlation different from entanglement are put forward too. For instance it had been understood [12] that quantum discord was undesignedly used and played an essential role in quantum computation [13] before its original presentation; quantum discords were responsible for some quantum implementations such as state merging [14] remote state preparation [15] assisted state discrimination [16] quantum phase transition [17] and quantum Grover research algorithm without entanglement [18 19] etc. Recently quantum correlation different from entanglement has been attracting much attention and become a hot topic [20 39] in quantum information processing. In 2010 Modi et al. [3] put forward a unified view of correlation for multipartite systems. They defined some quantum states associated with the concerned state and made use of relative entropies between relevant states to characterize various correlations. In their correlation picture quantum correlation is the relative entropy between the concerned state and its closest classical state. They called quantum correlation as quantum discord too. Nonetheless the comparability of their quantum discord with that proposed by Olliiver and Zurek [1] is indirect and surely not obvious. To obtain a direct comparability in the same year Dakić et al. [4] then presented a new quantum correlation measure named geometric discord for any bipartite qubit system. As a new quantum correlation measure recently geometric discord has attracted some researchers interests too. [ ] In this paper we will employ geometric discord as quantum correlation quantifier to study a family of separable states consisting of arbitrary mixtures of any two bi-qubit product pure states. As a matter of fact by far these states have been studied by some researchers. For instance Cen et al. [45] have already studied their quantum discord and entanglement of formation and derived analytic expressions. Nonetheless they mainly stressed the approach they presented in the paper. To reveal the concrete features of quantum discord amply Wang et al. [46] then utilized both analytic and numerical methods to systematically study quantum discords in the states. To our knowledge so far geometric discord has not been employed as quantifier to characterize quantum correlations in these states. In this paper we will treat this issue. On the other hand it is known that analytic quantum correlation ex- Supported by the National Natural Science Foundation of China (NNSFC) under Grant Nos and the Natural Science Foundation of Anhui Province under Grant No MA12 and the 211 Project of Anhui University zjzhang@ahu.edu.cn c 2015 Chinese Physical Society and IOP Publishing Ltd

2 440 Communications in Theoretical Physics Vol. 63 pressions are rarely accessed to generally. By far there are only a few of states with comparatively higher inherent symmetry whose quantum correlation can be expressed analytically. [51128] In this paper we will try to derive an analytic expression of geometric discords in our concerned states. Moreover we will amply analyze the captured geometric discords to exhibit their inherent features. The rest of this paper is organized as follows. In Sec. 2 first the inherent symmetry of local unitary transformation in the family states consisting of any mixture of a pair of arbitrary bi-qubit product pure states is revealed. Then an analytic expression of geometric discords in the family of bipartite separable qubit states is derived. In Sec. 3 all geometric discord values of our concerned states are numerically worked out with respect to the resultant analytic formula. Furthermore some discussions and concrete analyses on the captured geometric discords are made so that distinct features are exposed. Finally a concise summary is given in Sec Geometric Discord in any Mixture of a Pair of any Bi-qubit Product States Let us first introduce the family of bipartite separable qubit states which we concern in this paper. They can be written as ρ ab = q Φ 1 a Φ 1 Φ 2 b Φ 2 + (1 q) Φ 3 a Φ 3 Φ 4 b Φ 4 (1) where a and b stand for two qubits q (0 1) denotes the relative mixing weight of the pair of product states and Φ s are arbitrary single-qubit states. With the computational bases any single-qubit state can be written as Φ = e iζ (cosθ 0 + e iδ sin θ 1 ) where ζ [0 2π) δ [0 2π) and θ [0 π/2]. Complementarily one can see that ρ ab is actually a state determined by the nine parameters q θ i and δ i with i = Easily one can rewrite the expression (1) as (see appendix) where ρ ab = U a (θ 1 δ 1 λ 1 ω 1 )U b (θ 2 δ 2 λ 2 ω 2 ) ab (q α 1 α 2 )U b (θ 2 δ 2 λ 2 ω 2 )U a (θ 1 δ 1 λ 1 ω 1 ) (2) ( cosθ e i(δ+ω λ) ) sin θ U(θ δ λ ω) = e iδ sin θ e i(ω λ) (3) cosθ { sin(δi+2 δ i )tanθ i tan θ } i+2 λ i = arctan (4) 1 + cos(δ i+2 δ i )tanθ i tan θ i+2 { sin δi tan θ i cotθ i+2 sin δ i+2 ) } ω i = arctan (5) cosδ i tan θ i cotθ i+2 cosδ i+2 ab (q α 1 α 2 ) = q 0 a 0 0 b 0 + (1 q) ϕ(α 1 ) a ϕ(α 1 ) ϕ(α 2 ) b ϕ(α 2 ) (6) ϕ(α i ) = α i α i 1 (7) α i = cos 2 (θ i θ i+2 ) sin2θ i sin 2θ i+2 sin 2 δ i δ i+2. (8) 2 Easily one can verify α i [0 1]. Obviously the kernel state ab (q α 1 α 2 ) is a three-parameter state. It associates with the concerned nine-parameter state ρ ab through local unitary operations. If a nine-parameter state ρ ab is given then the three-parameter kernel state ab (q α 1 α 2 ) is given too. This is the inherent local unitary transformation symmetry in the concerned states. Now let us derive geometric discords in our concerned states ρ ab. Geometric discord presented by Dakić et al. [4] in 2010 as quantum correlation quantifier for any bipartite qubit system is defined as Q G [ρ ab ] = min {χ ab } ρ ab χ ab 2 (9) where the minimum is taken over all zero-discord states χ ab of ρ ab and ρ ab χ ab 2 = tr(ρ ab χ ab ) 2 is the square norm in the Hilbert-Schmidt space. Naturally the geometric discord of ab is Q G [ ab (q α 1 α 2 )] = min {χ ab } ab χ ab 2 (10) where {χ ab } denotes all zero-discord states of ab. According to the definition of quantum discord it is very easy to verify that U a U b χ ab U b U a are also zero-discord states of ab. This means that tr(ρ χ) 2 is unchanged under local unitary transformations and therefore the equality Q G (ρ ab ) = Q G ( ab ) holds. The equality actually indicates that geometric discord keeps unchanged if the state is converted by local unitary transformations. Because of this later we will directly consider the kernel state ab (q α 1 α 2 ) instead. In 2010 Luo et al. [5] put forward an approach to analytically obtain geometric discord in any bipartite state. For our concerned bi-qubit states their geometric discord formula can be derived as follows. First expand the kernel states

3 No. 4 Communications in Theoretical Physics 441 ab (q α 1 α 2 ) as ab (q α 1 α 2 ) = ij c ij (q α 1 α 2 )X (i) a X (j) b (11) where X s can be expressed in computational bases as X (0) = 1 ( ) 1 0 X (1) = 1 ( ) 0 1 X (2) = and c ij (q α 1 α 2 ) = tr ab ( ab X (i) a ( 0 i 2 i 0 ) X (3) = 1 ( ) 1 0 (12) X (j) b ). Then let C = (c ij ) and D = (d ki ) where d ki = tr( k k X (i) ) with k = 1 2 and i = and { k } forms a set of orthonormal bases for a qubit. Without loss of generality { k } can be parameterized as {cosφ 0 + e iη sin φ 1 sin φ 0 e iη cosφ 1 } where 0 φ π/2 and 0 θ 2π. According to the definitions of C and D after some tedious deductions one can get C = 1 2p p 2 α 2 (1 α 2 ) 0 p 2(1 α 2 ) 2 α 1 (1 α 1 ) 4 α 1 (1 α 1 )α 2 (1 α 2 ) 0 2 α 1 (1 α 1 )(2α 2 1) p 2(1 α 1 ) 2 α 2 (1 α 2 )(2α 1 1) 0 p 2[α 1 (1 α 2 ) + α 2 (1 α 1 )] D = 1 ( ) 1 sin2φcosη sin 2φsinη cos2φ (14) 2 1 sin 2φcosη sin2φsinη cos2φ (13) where p = 1/(1 q). Finally by virtue of Luo et al. s study [5] the geometric discords in the kernel states are formally expressed as Q G [ ab (q α 1 α 2 )] = tr(cc t ) max D {tr(dcct D t )} = 1/2 q(1 q)(1 2α 1 α 2 + α 2 ) where t represents transposition and ξ s are defined as max D {ξ 1 sin 2 2φcos 2 η + ξ 2 cos 2 2φ + ξ 3 sin2φcosη cos2φ} (15) ξ 1 = 2(1 q) 2 α 1 (1 α 1 ) ξ 2 = 1/2 (1 q)(1 2α 1 α 2 + α 2 ) + (1 q) 2 [α 2 1(1 α 2 ) + (1 + α 2 )(1 α 1 ) 2 ] ξ 3 = 2(1 q) α 1 (1 α 1 )[q(1 2α 1 + α 2 ) + 2α 1 1]. (16) To really obtain geometric discord in a given kernel state the minimization in formula (15) needs to be executed with respect to the two measuring parameters φ and η. In this case one must work out the extreme points by solving the derivative equations tr(dcc t D t ) = 2(ξ 1 cos 2 η ξ 2 )sin 4φ + 2ξ 3 cos4φcosη = 0 φ tr(dcc t D t ) = (2ξ 1 sin 2 2φcosη + ξ 3 sin2φcos2φ)sin η = 0. (17) η After complicated analyses and some calculations one can get the extreme points (η e φ e ) i.e. ( arctan 2ξ ) ( 3 π 1 ξ 2 ξ 1 4 arctan 2ξ ) ( 3 π ) ( 3π ) ( π ξ 1 ξ π ) ( 3π 4 2 π ) ( π 4 2 π ) 2 ( 3π 2 π 2 ). (18) Substituting these extreme points into Eq. (15) and comparing the corresponding geometric discord values one can easily get the minimal points (η m φ m ) for geometric discord i.e. ( arctan 2ξ ) ( 3 π 1 ξ 2 ξ 1 4 arctan 2ξ ) 3. (19) ξ 1 ξ 2 Consequently the analytical expression of geometric discord in the state ab is Q G [ ab (q α 1 α 2 )] = 1/2 q(1 q)(1 2α 1 α 2 + α 2 ) 1/2[(ξ 1 + ξ 2 ) + (ξ 2 ξ 1 )cos4φ m + ξ 3 sin 4φ m ]. (20) Obviously the captured geometric discord is a function of q α 1 and α 2. 3 Analyses and Discussions In the last section we have derived the analytic expression of geometric discords in the family of bipartite separable qubit states ρ ab. By virtue of formula (20) we have calculated the concrete geometric discord values in all the concerned states by numerical computations. Some representatives are shown in Figs Now let us make some discussions and analyses about them. From the figures one can see the following distinct features.

4 442 Communications in Theoretical Physics Vol. 63 Fig. 1 Geometric discord as a function of q and α 1 with α 2 = 1/4 1/2 3/4. Fig. 2 Geometric discord as a function of q and α 2 with α 2 = 1/4 1/2 3/4. (i) Geometric discords are symmetric about q = 1/2. From Figs. 1 and 2 one can see that Q G [ ab (q α 1 α 2 )] = Q G [ ab (1 q α 1 α 2 )]. It seems that the equality holds for all α 1 s and α 2 s. Actually one can easily prove the equality by substituting q and 1 q respectively into Eq. (20). Hence the equality holds for all α 1 s and α 2 s indeed. This indicates that the symmetry is a general feature. Its underlying physics can be explained as follows. One can easily verify that W a (α 1 )W b (α 2 ) ab (q α 1 α 2 )W a(α 1 )W b (α 2) = ab (1 q α 1 α 2 ) (21) where the local single-qubit operation W takes the following matrix form ( ) α 1 α W(α) =. (22) 1 α α As stressed in the last section geometric discord is invariant if the state is converted by local unitary transformations. Obviously ab (q α 1 α 2 ) is associated with ab (1 q α 1 α 2 ) via local unitary transformations. As a result geometric discords in both states should be equal i.e. Q G [ ab (q α 1 α 2 )] = Q G [ ab (1 q α 1 α 2 )]. Owing to such symmetry later we will restrict our analyses and discussions on the captured geometric discords within the scope q (0 1/2]. (ii) Geometric discord monotonically increases with the increasing q in the region [0 1/2]. Obviously with respect to the symmetry feature it should monotonically decrease with the increasing q in the region [1/2 1]. From Fig. 1 one can see that geometric discord monotonically increases with the increasing q in the region [0 1/2] provided that α 1 is given. Similarly it monotonically increases with the increasing q in the region [0 1/2] in the case that α 2 is given as can be easily seen in Fig. 2. All these mean that if the two bi-qubit product states are given then the more averagely they are mixed the bigger geometric discord the mixed state owns. Particularly geometric discord reaches its maximal value when two given product states are mixed with the equal weight 1/2. (iii) Geometric discord first increases and then decreases with the increasing α 1 in the case of any given mixing weight q. One can easily see this feature from each subgraph of Fig. 1. Nonetheless the transition points in three subgraphs are obviously different. Both the transition point s value (i.e. α 1 s value) and its corresponding geometric discord become smaller with the increasing α 2.

5 No. 4 Communications in Theoretical Physics 443 (iv) Geometric discord always decreases with the increasing α 2 for any given mixing weight q. One can easily see this feature from each subgraph of Fig. 2. From items (iii) and (iv) one can see that the dependence relationships of geometric discord on α 1 and α 2 are distinctly different in the qualitative aspect. In fact they are also obviously different in the quantitative aspect as can be seen from Figs. 1 and 2. The essential reason of the differences originates from the retrieval of geometric discord. Note that geometric discord in a state is defined as the minimal one among the Hilbert-Schmidt distances between the state and its zero-discord states. The zerodiscord states are actually posterior states corresponding to single-partite orthogonal measurements on the original state. In the analytic expression of geometric discord given by formula (20) the measurement on partite a is actually assumed in priori. This preassumption is responsible for the difference of dependence relationships on α 1 and α 2. (v) The maximal geometric discord among those in this family of states is 1/8. Integrating items (ii) (iv) and Figs. 1 2 one can easily conclude that the maximal geometric discord among those in this family of states occurs at q = 1/2 α 1 = 1/2 and α 2 = 0. Substituting q = 1/2 α 1 = 1/2 and α 2 = 0 into formula (20) one can get the maximal geometric discord equal to 1/8 which is the upper limit of geometric discord in the concerned states. Finally we want to stress that although we have only obtained an analytical expression of geometric discords in the kernel states ab (q α 1 α 2 ) and revealed some relevant features about the captured geometric discords actually both the expression and features are applicable for the nine-parameter states ρ ab we concern. As mentioned before the three-parameter kernel states ab (q α 1 α 2 ) are clearly associated with the concerned nine-parameter state ρ ab through local unitary operations. Alternatively the three-parameter kernel states ab (q α 1 α 2 ) are completely determined by the nine-parameter states ρ ab with the explicit correspondence relationships given by formula (8). Because of this we say the expression and features are applicable for the nine-parameter states ρ ab. 4 Summary To summarize in this paper we have studied geometric discords in a family of states comprising any mixture of a pair of arbitrary bi-qubit product pure states and characterized by nine parameters. We have revealed the inherent symmetry in the family of states about local unitary transformations and thus extracted corresponding three-parameter kernel states instead of the initial nineparameter states for the geometric discord study. Subsequently we have explicitly derived an analytic expression of geometric discords in the kernel states. Furthermore we have made some analyses and discussions about the captured geometric discords and revealed some distinct features. We find that the more averagely the two bi-qubit product states are mixed the bigger geometric discord the mixed state owns. Moreover we have shown the monotonic relationships of geometric discord with different parameters. At last we have simply demonstrated that both the analytic expression and the distinct features are applicable for the states we concerned initially. Appendix Brief Proof for Eq. (2) U a (θ 1 δ 1 λ 1 ω 1 ) and U b (θ 2 δ 2 λ 2 ω 2 ) are defined by Eqs. (3) (5). Performing them on state ρ ab expressed in Eq. (1) after some tedious deductions one can easily get U a(θ 1 δ 1 λ 1 ω 1 )U b (θ 2 δ 2 λ 2 ω 2 )ρ ab U a (θ 1 δ 1 λ 1 ω 1 )U b (θ 2 δ 2 λ 2 ω 2 ) = q 0 a 0 0 b 0 + (1 q)u a(θ 1 δ 1 λ 1 ω 1 )U b (θ 2 δ 2 λ 2 ω 2 ) Φ 3 a Φ 3 Φ 4 b Φ 4 U a (θ 1 δ 1 λ 1 ω 1 )U b (θ 2 δ 2 λ 2 ω 2 ). (23) Using Φ = e iζ (cosθ 0 + e iδ sinθ 1 ) one can further get U a (θ 1 δ 1 λ 1 ω 1 )U b (θ 2 δ 2 λ 2 ω 2 ) Φ 3 a Φ 4 b = e i(λ1+λ2) ( α 1 0 a + 1 α 1 1 a )( α 2 0 b + 1 α 2 1 b ). (24) Correspondingly one is readily to get U a (θ 1 δ 1 λ 1 ω 1 )U b (θ 2 δ 2 λ 2 ω 2 )ρ ab U a (θ 1 δ 1 λ 1 ω 1 )U b (θ 2 δ 2 λ 2 ω 2 ) = q 0 a 0 0 b 0 + (1 q) ϕ(α 1 ) a ϕ(α 1 ) ϕ(α 2 ) b ϕ(α 2 ) ab (q α 1 α 2 ) (25) where ϕ(α i ) = α i α i 1 is given in Eq. (7). This indicates ρ ab = U a (θ 1 δ 1 λ 1 ω 1 )U b (θ 2 δ 2 λ 2 ω 2 ) ab (q α 1 α 2 )U b (θ 2 δ 2 λ 2 ω 2 )U a(θ 1 δ 1 λ 1 ω 1 ). (26) References [1] H. Ollivier and W.H. Zurek Phys. Rev. Lett. 88 (2001) [2] S.L. Luo Phys. Rev. A 77 (2008) [3] K. Modi T. Paterek W. Son V. Vedral and M. Williamson Phys. Rev. Lett. 104 (2010)

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