Quantum Discord in Two-Qubit System Constructed from the Yang Baxter Equation

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1 Commun. Theor. Phys. 61 (014) Vol. 61, No. 3, March 1, 014 Quantum Discord in Two-Qubit System Constructed from the Yang Baxter Equation GOU Li-Dan ( ), 1 WANG Xiao-Qian ( ¼), 1, XU Yu-Mei (Å Ö), and SUN Yuan-Yuan (êûû) 1,3 1 Department of Physics, Changchun University of Science and Technology, Changchun 1300, China Department of Admissions, Weifang Nursing Vocational College, Weifang 6500, China 3 Combat Command Department, Aviation University of Air Force, Changchun 1300, China (Received July 10, 013; revised manuscript received October 8, 013) Abstract Quantum correlations among parts of a composite quantum system are a fundamental resource for several applications in quantum information. In general, quantum discord can measure quantum correlations. In that way, we investigate the quantum discord of the two-qubit system constructed from the Yang Baxter Equation. The density matrix of this system is generated through the unitary Yang Baxter matrix R. The analytical expression and numerical result of quantum discord and geometric measure of quantum discord are obtained for the Yang Baxter system. These results show that quantum discord and geometric measure of quantum discord are only connect with the parameter θ, which is the important spectral parameter in Yang Baxter equation. PACS numbers: a, fd Key words: quantum correlations, quantum discord, Yang Baxter equation 1 Introduction Nowadays, quantum correlations have attracted much attention since it plays a crucial role in quantum information. Quantum discord (QD), a new measure of quantum correlations, has been proposed based on the information theoretic concept of mutual information. [1 3] QD is quantified by the difference between two quantum extensions of the classical mutual information. According to this conception, QD can describe the non-classical correlations besides entanglement. [4] Recently, QD have been researched in different systems, e.g., low dimensional spin models, [5 9] open quantum system, [10 13] biological, [14] and relativistic system. [15] Unfortunately, QD is difficult to calculate and it is usually based on a numerical maximization procedure. There are few analytical expressions of QD in some special cases. To avoid this difficulty, Daki c et al. [16] introduced the geometric measure of quantum discord (GMQD) to describe the quantum correlations, which is based on the minimal Hilbert Schmidt distance between the given state and a zero-discord state. And GMQD is derived an explicit formula for any twoqubit state in 010. [17] There are also many discussion of GMQD in different systems. In the past decades, the Yang Baxter equation (YBE) plays the fundamental role in the theory of (1+1)- or -dimensional integrable quantum system, [18 19] and it is also closely related to some other fields, such as lowdimensional statistical models, chain models, and lowdimensional quantum field models. [0 1] Recently, the YBE and braiding operators have been introduced to the field of quantum information, and they also provide a novel way to study the quantum entanglement. [ 9] Reference [] has explored the role of the unitary solution of YBE in quantum information. It has been shown that the solution of YBE can be identified as the universal quantum gate. [ 4] It motivates a novel way to study quantum entanglement, which is based on the theory of braiding operators, as well as YBE. Zhang, Kauffman, and Ge [4] first presented that the Bell matrix, generating two-qubit entangled states, has been recognized as a unitary braid transformation. After that more and more researches show that any pure two-qubit entangled states can be achieved by a universal Yang Baxter Rmatrix. [6 9] Recent researches show that the entanglement describes only a part of quantum correlations but not all of them. Yet QD can go beyond the concept of entanglement and obtain the quantumness of the correlations between two parts of a system. QD may be used as a powerful tool to study quantum correlations existing in the correlated quantum states. Motivated by this, we are interested in the quantum correlations of the Yang Baxter system. We calculate and analyse QD and GMQD of the two-qubit system constructed from YBE. Thus we can know well the quantum correlations of the Yang Baxter system. It may disclose the new scenarios for YBE. Supported by the National Natural Science Foundation of China under Grant Nos and and the CUST Foundation for Young Scholars under Grant No. XQNJJ xqwang1@163.com c 013 Chinese Physical Society and IOP Publishing Ltd

2 350 Communications in Theoretical Physics Vol. 61 The paper is organized as follows. In Sec., we introduce the quantum discord and geometric measure of quantum discord. In Sec. 3, we discuss the matrix presentation of braid group, and obtain the solution R-matrix of YBE. According to the solution of YBE, we can get four new states. Then we use the new states to construct the density matrix of the two-qubit system. We can separately get the analytical expressions of QD and GMQD about the pure states and the numerical result of QD about the mixed states. In the section, we find that QD and GMQD are only connected with the one parameter in the YBE. Conclusions are then presented in Sec. 4. Quantum Discord and Geometric Measure of Quantum Discord In this section we first give a brief introduction of QD. We consider a bipartite state ρ in a Hilbert space H (A) H (B). The quantum discord δ(ρ) is defined as the difference between the total correlations and the classical correlations, and the form is [1 ] δ(ρ) = I(ρ) C(ρ), (1) where I(ρ) represents the total correlations of the bipartite state ρ. And it can be written as I(ρ) = S(ρ A ) + S(ρ B ) S(ρ), () where ρ A(B) = Tr B(A) ρ is the reduced density matrix for subsystem A(B). S(ρ) = Tr(ρ log (ρ)) is the Von Neumann entropy. The other quantity, C(ρ) is interpreted as a measure of the classical correlations of the bipartite state ρ. We can get the amount of classical correlations, which is expressed by the mutual information and is obtained by adopting the least disturbing measurement. C(ρ) is defined as C(ρ) = max {B k } (S(ρ a) S(ρ {B k })), (3) where S(ρ {B k }) is quantum conditional entropy, the form is S(ρ {B k }) = p k S(ρ k ), (4) k where {B k } means a measurement performed locally only on party B. ρ k may be considered as a conditional density operator, which is conditioned on the measurement outcome labeled by k. The form can be written as ρ k = 1 p k (I B k )ρ(i B k ), (5) with the probability p k = Tr(I B k )ρ(i B k ). Here I means the identity operator for party A. Since the calculation of quantum discord is based on maximization procedure, there are few analytical expressions in many quantum systems. To avoid this difficulty, Dakic, et al. introduced geometric measure of quantum discord (GMQD), which is defined as the minimum Hilbert-Schmidt distance between the given state and a zero-discord state. It can be written as [16 17] D(ρ) = min χ ρ χ, (6) where the geometric quantity ρ χ = Tr(ρ χ) is the square of Hilbert Schmidt norm of Hermitian operators. For any two-qubit state ρ = 1 3 R αβ σ α σ β. (7) 4 α,β=0 Here σ 1,,3 are the Pauli matrices σ x,y,z, and σ 0 is the unit matrix. R αβ can be expressed as R αβ = Tr[σ α σ β ρ]. (8) It takes the following matrix form ( 1 b T ) R =. (9) a τ Here the superscript T denotes transpose of vectors or matrices. Then a = (a 1, a, a 3 ) T and b = (b 1, b, b 3 ) T are column vectors, and τ is a 3 3 matrix. Then the geometric measure of quantum discord is evaluated as D(ρ) = 1 4 ( b + τ λ max ), (10) where λ max is the largest eigenvalue of the matrix b b T + ττ T. 3 Quantum Discord and Geometric Measure of Quantum Discord in Yang Baxter System In Ref. [8], the authors presented a new S-matrix and got a unitary R-matrix. It is shown that the unitary Rmatrix can generate the new states. The S-matrix, a solution of the braid relation, has the following form 0 e iφ i e iφ 0 S = 1 e iφ 0 0 e iφ i e iφ 0 0 i e iφ, (11) 0 e iφ i e iφ 0 where φ is real. We can verify that S-matrix is unitary. Next we derive a unitary matrix R from the S-matrix via the Yang Baxterization approach. In this work, we are interested in the following rational YBE R i (u) R i+1 (u+v) R i (v) = R i+1 (v) R i (u+v) R i+1 (u). (1) The spectral parameters u and v usually play an important role, which are related to the one-dimensional momentum in some typical models. By introducing a new variable θ with cosθ = u/ 1 + u and sinθ = 1/ 1 + u, the R(u) can be written as R(θ, φ) = sin θi + i cosθs, (13) here I means the identity operator. There are two parameters θ and φ in the unitary matrix R(θ, φ). θ is the spectral parameter, which is introduced from Yang Baxterization,

3 No. 3 Communications in Theoretical Physics 351 while φ is the phase factor, which is introduced from the braid relation. We choose { 00, 01, 10, 11 } as the bases. When the unitary Yang Baxter matrix R(θ, φ) acts on these bases, the bases can be changed to the new states, which can be written as Ψ 1 = sin θ 00 + i cosθ e iφ cosθ e iφ 10, Ψ = i cosθ e iφ 00 + sinθ 01 θ = 0, π, QD gets the maximum value of 1. And when θ = π/, QD gets the minimum value of 0. In the vicinity of θ = π/, the trend of QD is a little plane, because of the logarithm in the analytical expression. When θ varies from 0 to π, the four states possess continuous quantum correlations determined by θ. When θ = 0, π, which corresponds to the Bell states, the QD reaches the maximum value. That is to say, the action of R-matrix results in the two-qubit maximally quantum correlation states. At the case of θ = π/, R corresponds to an identity operation with no quantum correlations at all. + i cosθ e iφ 11, Ψ 3 = 1 cosθ e iφ 00 + sin θ cosθ e iφ 11, Ψ 4 = i cosθ e iφ 01 1 cosθ e iφ 10 + sin θ 11. (14) In this paper we consider the quantum correlations of these states in Eq. (14), which are transformed by the unitary Yang Baxter matrix R. The density matrices are constructed by using the new states, namely ρ i = Ψ i Ψ, (i = 1,, 3, 4). Fig. Quantum discord is plotted versus θ. Fig. 1 Quantum discord is plotted versus θ and φ. We give the numerical result of QD with different θ and φ. From Fig. 1, we can find that the QD is not connect with parameter φ. We give the exact expression of QD. The exact expression of QD can be given as the following form δ(ρ i ) = 1 1 cos 4 θ cos 4 θ log 1 1 cos 4 θ cos log 4 θ (i = 1,, 3, 4), (15) where δ(ρ i ) (i = 1,, 3, 4) denotes the i-th state s QD. We can choose QD as a function of θ, and use graphics to portray the behavior of it in Fig.. We find that when Fig. 3 Geometric measure of quantum discord is plotted versus θ. According to Eqs. (6) (10), the geometric measure of quantum discord of this Yang Baxter system can be evaluated as D(ρ i ) = 1 cos θ (i = 1,, 3, 4), (16) where D(ρ i ) (i = 1,, 3, 4) denotes the i-th state s GMQD. We can choose GMQD as a function of θ, and use graphics to portray the behavior of it in Fig. 3. It is found that when θ = 0, π, GMQD gets the maximum value of 0.5. And when θ = π/, GMQD gets the minimum value

4 35 Communications in Theoretical Physics Vol. 61 of 0. GMQD ranges always from 0 to 0.5. In this plot, comparing Figs. and 3, we can find the trends of QD and GMQD are similar extremely. When θ = 0, π, which corresponds to the Bell states, GMQD reaches the maximum value. That is to say, the action of the R-matrix results in the two-qubit maximally quantum correlation states. When θ is by π/, the trend of the curve of QD is more plane than GMQD, because of the logarithm in the analytical expressions. Therefore we may use the geometric measure of quantum discord instead of quantum discord, and GMQD can also be used to measure quantum correlations. It is worth to note that QD and GMQD only depend on the θ, which is the spectral parameter. However they do not depend on the parameter φ, which is the phase factor. This tells us that the effect of φ can be reduced by local unitary operation. It is noticed that these states in Eq. (14) are all pure states. We analyze QD and GMQD of these pure states as described above. Next we consider QD of the mixed states in Yang Baxter system. The density matrices are constructed by using the states in Eq. (14), namely ρ = p 1 Ψ i Ψ + p Ψ j Ψ, (i, j = 1,, 3, 4; i j), where p 1, [0, 1] and p 1 + p = 1. We can propose the form of the density matrices as [30] ρ = sin αρ i + cos αρ j, (α [0, π/]) (17) where ρ i = Ψ i Ψ and ρ j = Ψ j Ψ, (i, j = 1,, 3, 4; i j). The state ρ has rank two, and its QD reflects the degree of the correlations between the states of ρ i and ρ j. Fig. 4 Quantum discord of the mixed states is plotted versus θ and α. Fig. 5 Quantum discord of the mixed states is plotted versus θ for α = 0 (a), α = π/6 (b), α = π/4 (c), and α = π/3 (d). By calculating the QD of the mixed states, we find that it does not also depend on the parameter φ and plot the curve of QD with different θ and α in Fig. 4. After that we can choose QD as a function of θ when α has some specific values (see Fig. 5). On the one hand, Fig. 5 shows that QD gets the maximum value of 1 for α = 0 (a), 0.5 for α = π/6, π/3 (b)(d), and 0. for α = π/4 (c). This means that when α = π/4, the degree of the correlations between the states of ρ i and ρ j reaches to the minimum. When α = π/6, π/3, the correlations of them increase until α = 0, which is the maximal correlations of the pure state. On the other hand, if θ = π/, then QD gets the minimum value of 0 regardless of the value of α. In Fig. 5(a), when θ = 0, π, QD reaches the maximum value. Figures 5(b), 5(c), and 5(d) show that the point of θ, corresponding to the maximum value of QD, tends

5 No. 3 Communications in Theoretical Physics 353 toward π/ gradually. This means that the YBE spectral parameter θ has a marked impact on the degree of the correlations between the states of ρ i and ρ j. 4 Conclusion In this paper, based on these states by using the unitary Yang-Baxter matrix R, we construct the density matrices and calculate the QD and GMQD in the twoqubit system constructed from the Yang Baxter equation. Firstly, we give the analytical expressions of QD and GMQD about the pure states, and compare the properties of QD with that of GMQD. Remarkably, we find that QD and GMQD are all connected with the parameter θ, which is the important spectral parameter in the Yang Baxter equation. And they all get the minimum value at the point θ = π/ and the maximum value at the point θ = 0, π, which corresponds to the Bell states. Secondly, we get the numerical result of QD about the mixed states. It is also related to the parameter θ. We have investigated that the coefficient α affects to the degree of the correlations between the states of ρ i and ρ j. Morever QD and GMQD do not depend on the parameter φ, which is the phase factor. That is to say, the quantum correlations of this Yang Baxter system can not be affected by this phase factor φ. In short, we can obtain the states possessing the arbitrary degree of correlations through the Yang Baxter matrix R. References [1] H. Olivier and W.H. Zurek, Phys. Rev. Lett. 88 (001) [] W.H. Zurek, Rev. Mod. Phys. 75 (003) 715. [3] L. Henderson and V. Vedral, J. Phys. A 34 (001) [4] A. Datta, A. Shaji, and C.M. Caves, Phys. Rev. Lett. 100 (008) [5] R. Dillenschneider, Phys. Rev. B 78 (008) [6] M.S. Sarandy, Phys. Rev. A 80 (009) [7] T. Werlang and G. Rigolin, Phys. Rev. A 81 (010) [8] Y.X. Chen and S.W. Li, Phys. Rev. A 81 (010) [9] X.M. Lu, J. Ma, Z.J. Xi, and X.G. Wang, Phys. Rev. A 83 (011) 137; X.L. Yin, Z.J. Xi, X.M. Lu, Z. Sun, and X.G. Wang, J. Phys. B 44 (011) 4550; Z.J. Xi, X.M. Lu, and X.G. Wang, J. Phys. A 44 (011) [10] J. Maziero, L.C. Céleri, R.M. Serra, and V. Vedral, Phys. Rev. A 80 (009) 04410; J. Maziero, T. Werlang, F.F. Fanchini, L.C. Céleri, and R.M. Serra, Phys. Rev. A 81 (010) [11] A. Shabani and D.A. Lidar, Phys. Rev. Lett. 10 (009) [1] T. Werlang, S. Souza, F.F. Fanchini, and C.J. Villas- Boas, Phys. Rev. A 80 (009) 04103; F.F. Fanchini, T. Werlang, C.A. Brasil, L.G.E. Arruda, and A.O. Caldeira, Phys. Rev. A 81 (010) [13] K. Modi, T. Paterek, W. Son, V. Vedral, and M. Williamson, Phys. Rev. Lett. 104 (010) [14] K. Bradler, M.M. Wilde, S. Vinjanampathy, and D.B. Uskov, arxiv: [15] A. Datta, Phys. Rev. A 80 (009)05304; L.C. Celeri, A.G.S. Landulfo, R.M. Serra, and G.E.A. Matsas, arxiv: [16] B. Dakic, V. Vedral, and C. Brukner, Phys. Rev. Lett. 105 (010) [17] S.L. Luo and S.S. Fu, Phys. Rev. A 8 (010) [18] C.N. Yang, Phys. Rev. Lett. 19 (1967) 131; C.N. Yang, Phys. Rev. 168 (1968) 190. [19] R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York (198); R.J. Baxter, Ann. Phys. 70 (197) 193. [0] L.D. Faddev, Integrable Models in (1+1)-Dimensional Qumfum Field Theories, Les Houches Lecture (198), Amsterdam, Elsvier (1984) 536. [1] P.P. Kulish and E.K. Sklyanin, Lecture Notes in Physics, Springer, Berlin 151 (198) 61. [] L.H. Kauffman and S.J. Lomonaco Jr, New J. Phys. 6 (004) 134. [3] J.M. Franko, E.C. Rowell, and Z. Wang, J. Knot Theory Ramif. 15 (006) 413. [4] Y. Zhang, L.H. Kauffman, and M.L. Ge, Int. J. Quant. Inf. 3 (005) 669. [5] Y. Zhang and M.L. Ge, Quant. Inf. Proc. 3 (007) 363. [6] J.L. Chen, K. Xue, and M.L. Ge, Phys. Rev. A 76 (007) [7] J.L. Chen, K. Xue, and M.L. Ge, Ann. Phys. 33 (008) 614. [8] G.C. Wang, K. Xue, et al., J. Phys. A: Math. 4 (009) [9] L.D. Gou, K. Xue, et al, Commun. Theor. Phys. 55 (011) 63. [30] F.L. Zhang, J.L. Chen, L.C. Kwek, and V. Vedral, Scientific Reports 3 (013) 134.