Quantum Correlation in Matrix Product States of One-Dimensional Spin Chains

Transcription

1 Commun. Theor. Phys. 6 (015) Vol. 6, No. 3, September 1, 015 Quantum Correlation in Matrix Product States of One-Dimensional Spin Chains ZHU Jing-Min ( ) College of Optoelectronics Technology, Chengdu University of Information Technology, Chengdu 6105, China (Received January 8, 015; revised manuscript received July 6, 015) Abstract For our proposed composite parity-conserved matrix product state (MPS), if only a spin block length is larger than 1, any two such spin blocks have correlation including classical correlation and quantum correlation. Both the total correlation and the classical correlation become larger than that in any subcomponent; while the quantum correlations of the two nearest-neighbor spin blocks and the two next-nearest-neighbor spin blocks become smaller and for other conditions the quantum correlation becomes larger, i.e., the increase or the production of the long-range quantum correlation is at the cost of reducing the short-range quantum correlation, which deserves to be investigated in the future; and the ration of the quantum correlation to the total correlation monotonically decreases to a steady value as the spacing spin length increasing. PACS numbers: Jm, Mn Key words: matrix product state (MPS), long-range quantum correlation, geometry quantum discord (GQD) Quantum systems not only can have classical correlation but also can have quantum correlation due to the quantum states described by the superposition of probability amplitude. The quantum correlation is a key source in quantum information which is prior to classical information. The quantum correlation properties of quantum spin chains can be used to realize many kinds of quantum information processing. [1 5] Hence the investigation of quantum correlation of quantum spin chains is undoubtedly an intensive research subject. The seldomness existence of the ground state analytic solution of the system Hamiltonian one time made the study of quantum correlation of quantum many-body systems run into trouble. Via the concepts and methods in quantum information science, the study of quantum many-body systems fended off the aforementioned difficulty. For example, MPSs is a powerful numerical computational simulation platform for studying one-dimensional spin chains by using quantum information approach which deals primarily with the quantum many-body state and the corresponding Hamiltonian can be constructed such that the given state exists as its ground state. [6 13] In this paper, we investigate the quantum correlation properties of quantum spin chains by using MPSs. First let us begin with a one-dimensional translation invariant MPS: Ψ = 1 Tr E N d i 1,i,...,i N =1 Tr (A i1 A i A i N ) i 1 i i N, (1) where d represents the freedom of one spin and the set of D D matrices {A 1, A,..., A d } parameterize the correlation of the system with the dimension D d N/. [9] In the normalized factor E = d i (Ai ) A i is the so-called transfer matrix where the symbol denotes complex conjugation. When (A i ) = A i the system has time reversal symmetry. Here we consider the composite parity conserved MPS Ψ with the form where { A 1 s = [ ] 1 0, A s = 0 γ {A i = A i s A i r, i = 1,, 3}, [ ] 0 1, A 3 s = 0 0 [ ] } 0 0, γ 0, ±1 γ 0 represent the parity non-conserved MPS Ψ s and {A i r = (A i s) T, i = 1,, 3} denote its parity conjugated MPS Ψ r = P N Ψ s. [1 15] The largest eigenvalue of the transfer matrix E of the parity conserved MPS is λ max = λ s = λ r = 1+γ with their corresponding right (left) eigenvectors λ R(L) s and λr R(L) respectively corresponding to the subcomponent transfer matrices E s and E r. For an adjacent l-spin block B l, its density matrix ρ l is expressed as ρ i1 i l,j 1 j l = Tr (Ai1 A i l E N l ) Tr E N, () Supported by the National Natural Science Foundation of China under Grant No and the Major Natural Science Foundation of the Educational Department of Sichuan Province under Grant No. 1ZA zhjm-6@163.com c 015 Chinese Physical Society and IOP Publishing Ltd

2 No. 3 Communications in Theoretical Physics 357 in the thermodynamic limit, which reduces to ρ i1 i l,j 1 j l = λl s A i1 A i l λ R s + λ L r A i1 A i l λ R r λ l s + λ l r = 1 (ρs i 1 i l,j 1 j l + ρ r i 1 i l,j 1 j l ) = 1 (ρs i 1 i l,j 1 j l + ρ s i l i 1,j l j 1 ). (3) For any two spin blocks B l and B l with spacing spin number n, their correlation density matrix reads C n [ρ l, ρ l ] = ρ n l,l ρ lρ l, () in the thermodynamic limit and n taking the thermodynamic limit, which reduces to C [ρ l, ρ l ] = (ρs l ρr l )(ρs l ρr l ). (5) It shows that the proposed composite parity-conserved MPS has correlation for any two spin blocks where their spin blocks length l, l > 1, due to the parity conjugated parity-absent MPS pair equally existing. In order to have some intuitive understanding of the aforementioned correlation in the proposed MPS, we study the correlation properties of local physical quantities. For a local physical operator of l adjacent spins Ô [1,l] = Ô k+1 Ôk+l, its correlation function is obtained as C n [Ô[1,l] ] = Ô[1,l] Ô [n+1,n+l] Ô[1,l] = Tr (Ô[1,l] C n [ρ l, ρ l ]). (6) For simplify, we consider the two-body parity absent operator Jx k+1 (Jx k+ ), the correlation C n [(J x ) [1] (Jx) [] ] varying with the spacing spin number n for γ = (1 + ) is shown in Fig. 1. The correlation C n [(J x ) [1] (Jx) [] ] of the specified MPS Ψ described by the black-point curve monotonically approaches to from the value of as the spacing spin number n increasing, while the corresponding correlations C n [(J x ) [1] (Jx) [] ] of the parity conjugated MPS pair Ψ s and Ψ r respectively described by the dark-grey-point curve and the grey-point curve quickly decrease to zero respectively from the values of and It is readily seen that the correlation C n [(J x ) [1] (Jx) [] ] of the composite MPS is larger than that of any subcomponent and the specified system has long-range correlation. When n, the corresponding long-range correlation about Eq. (6) reduces to C [Ô[1,l] ] = Tr (Ô[1,l] C [ρ l, ρ l ]) the maximal value 1/18= consistent with the aforementioned conclusion. It is concluded that the local parity absent physical quantities of the proposed Ψ have long-range correlation due to the parity conjugated parity-absent MPS pair equally existing apart the special points γ = 0, ±1 where Ψ s = Ψ r having parity symmetry. Fig. 1 The correlation of the two-body parity nonconserved operator C n[(j x) [1] (Jx) [] ] for γ = 1 +, varying with the spacing spin number n. The correlation C n[(j x) [1] (Jx) [] of the state Ψ described by the blackpoint curve monotonically approaches to from the value of with the increase of the spacing spin length n, while the corresponding correlations of the parity conjugated MPS pair Ψ s and Ψ r respectively represented by the dark-grey-point curve and the greypoint curve quickly decrease to zero. = 1 ( Ô[1,l] s Ô[1,l] r ). (7) (J k+ x For the same two-body parity absent operator Jx k+1 ), its long-range correlation C [(J x ) [1] (Jx) [] ] varying with the dimensionless parameter γ is shown in Fig.. The long-range correlation being even function about γ is nonzero apart the three points γ = 0, ±1 and when γ = ±(1 ), ±(1+ ), the long-range correlation takes Fig. The long-range correlation of the two-body parity absent operator (J x) [1] (Jx) [], C [(J x) [1] (Jx) []] as a function of the dimensionless parameter γ. The long-range correlation being even function about γ is nonzero apart the three points γ = 0, ±1 and when γ = ±(1 ), ±(1 + ), the long-range correlation takes the maximal value of 1/18=

3 358 Communications in Theoretical Physics Vol. 6 The aforementioned facts inspire us to comprehensively and deeply understand the correlation properties possessed in the proposed system, in the following we shall respectively and quantitatively investigate the total correlation, quantum correlation and classical correlation of two two-spin blocks B and B varying with their spacing spin number n in detail. First let us study the total correlation of two two-spin blocks with their spacing spin number n. According to Refs. [16 18], the total correlation (TC) of the two two-spin blocks is expressed as TC = S(ρ B ) + S(ρ B ) S(ρ BB ), (8) where S(ρ B ) and S(ρ BB ) respectively describe the Von Neumann entropy of the two-spin block B reduced density matrix ρ B and the united two two-spin blocks reduced density matrix ρ BB. The TC of two two-spin blocks varying with their spacing spin number n for γ = 1 + is shown in Fig. 3. The TC in the proposed MPS Ψ described by the black-point curve monotonically approaches to from the value of with the increase of the spacing spin length n, while the corresponding TC of the parity conjugated MPS pair Ψ s and Ψ r represented by the dark-grey-point curve quickly decreases to zero from the value of 1.173, and the TC in the composite MPS is larger than that in any subcomponent. Fig. 3 The TC of two two-spin blocks varying with their spacing spin number n, for γ = 1 +. The TC in the MPS Ψ described by the black-point curve monotonically approaches to from the value of as the spacing spin length n increasing, while the corresponding TC of the parity conjugated MPS pair Ψ s and Ψ r represented by the dark-grey-point curve quickly decreases to zero from the value of Next, let us investigate the quantum correlation of the proposed system. Due to the inherent complexity and richness of a large number of interacting particles, there is no consistency about the classification, the depiction and the measure of quantum correlation of quantum manybody systems. Here we use the quantum correlation of two two-spin blocks varying with the spacing spin number to depict the quantum correlation of the proposed system. About measures of quantum correlation, there are many kinds of methods, such as the quantification characteristic function of quantum nonlocality, [19] Bell inequality, [0 1] the Von Neumann entanglement entropy, [ 8] averaged entropy, [9] quantum discord [16 18] and so on. Considering our system, we adopt geometry quantum discord [18,30] as a measure of quantum correlation. The geometry quantum discord (GQD) of the two two-spin blocks with the spacing spin length n is expressed as GQD = min Tr ρ BB K k B B B I B ρ BB I B, (9) k where K { k } is the set of Von Neumann measuring basis. The GQD of two two-spin blocks varying with their spacing spin length n is showed for γ = 1 + in Fig.. The GQD of the proposed MPS described by the black-point curve monotonically approaches to from the value of with the increase of the two blocks spacing spin length n, while the corresponding GQD of the parity conjugated MPS pair Ψ s and Ψ r described by the grey-point curve quickly decreases to zero from the value of The GQD of the composite MPS is larger than that of the subcomponent parity conjugated MPS pair when the spacing spin length n > 1, while the GQD of the two two-spin blocks with the spacing spin number 0 or 1 in the composite system is smaller than that in the any subcomponent. And we find that the ration of the two two-spin blocks GQD to the corresponding TC, GQD/TC monotonically decreases to from the value of with the increase of the two blocks spacing spin length n as shown in Fig. 5. The Classical correlation (CC) of two two-spin blocks CC = CC GQD varying with their spacing spin number n for γ = 1 + is shown in Fig. 6. The CC in the proposed MPS Ψ described by the blackpoint curve monotonically approaches to from the value of with the increase of the two blocks spacing spin length n, while the corresponding CC of the parity conjugated MPS pair Ψ s and Ψ r represented by the dark-grey-point curve quickly decreases to zero from the value of and the CC in the composite MPS is larger than that in any subcomponent. It is concluded that due to the two subcomponents equally existing, the proposed system has long-range quantum correlation and long-range classical correlation; and both the total correlation and the classical correlation become larger than that in any subcomponent; while the quantum correlation of the two nearest-neighbor spin blocks (n = 0) or the two next-nearest-neighbor (n = 1) spin blocks becomes smaller, for n > 1 the quantum correlation becomes larger; k

4 No. 3 Communications in Theoretical Physics 359 and the ration of the quantum correlation to the total correlation monotonically decreases to a steady value as the two spin blocks spacing length increasing. Fig. The GQD of two two-spin blocks varying with their spacing spin number n, for γ = 1+. The GQD in the proposed MPS Ψ described by the black-point curve monotonically approaches to from the value of as the two blocks spacing spin length increasing, while the corresponding GQD of the MPS pair Ψ s and Ψ r represented by the dark-grey-point curve quickly decreases to zero from the value of where one form of the MPS Ψ s Hamiltonian takes H s = ( γ +γ (S z(s i z ) ) + γ + ) (S i z) Sz Sz i i ( + γ (S i z) + γ γ (S i z) (S z ) ) + 1 ((S z ) Sz ) γ((s+) i Sz S+ + (S i ) S Sz ) + γ Si zsz + γ (S i +S i zs z S + S i zs i S + S z ) γ 3 (S i zs i +(S + ) + S i S i z(s ) ), (11) where S z = + +, S + = (S x + is y ), S = (S x is y ). Although the subcomponent Hamiltonian H s has no parity symmetry, the composite MPS Hamiltonian H has parity symmetry. Fig. 5 The ratio of two two-spin blocks quantum correlation to the corresponding total correlation, GQD/TC varying with the spacing spin number n, for γ = 1 +. The ratio of two two-spin blocks GQD to the corresponding TC in the proposed MPS Ψ monotonically approaches to from the value of as the two spin blocks spacing spin length increasing. Finally, we undertake the study of the dynamics of the specified MPS. According to Eq. (), the reduced density matrix of an l-adjacent spin block B l has at least l 3 D (l 3 > D ) zero eigenvectors. [9] We can construct a positive Hamiltonian H = k u KP k which is supported in that null space such that the proposed MPS exists as its ground state. Here the l takes the value of and the system Hamiltonian reads H = H s + H r = H s + P N H s, (10) Fig. 6 The CC of two two-spin blocks varying with their spacing spin number n, for γ = 1 +. The CC in the MPS Ψ described by the black-point curve approaches to from the value of as the two blocks spacing spin length increasing, while the corresponding CC of the conjugated MPS pair Ψ s and Ψ r represented by the dark-grey-point curve quickly decreases to zero from the value of Inclusion, MPSs provide a convenient platform for investigating the quantum correlation properties of quantum spin chains. Through the investigation of our proposed MPS, we think that the quantum correlation in the quantum many-body systems may be classified into two kinds, short-range quantum correlation and long-range quantum correlation, can be depicted by the quantum correlation of two spin blocks varying their spacing spin length and can be measured by geometry quantum discord. And we find that for MPSs, if the degeneracy of the largest eigenvalue of the transfer matrix is larger than 1

5 360 Communications in Theoretical Physics Vol. 6 and different eigenvector of the largest eigenvalue corresponds to different MPS, the system has long-range correlation; if not, the system has short-range correlation in general. For our proposed parity-conserved MPS, due to the parity conjugated parity-absent MPS pair equal existence, if only the spin block length is larger than 1, any such two spin blocks have quantum correlation and classical correlation. Both the total correlation and the classical correlation become larger than that of any subsystem; while the quantum correlation of the two nearest-neighbor spin blocks or the two next-nearest-neighbor spin blocks becomes smaller and for other conditions the quantum correlation becomes larger, that is to say, the long-range quantum correlation increase or production is at the cost of reducing the quantum short-range correlation, which deserves to be investigated in the future; and the ration of the quantum correlation to the total correlation monotonically decreases to a steady value with the increase of the two blocks spacing spin length. References [1] S.B. Zheng and G.C. Guo, Phys. Rev. Lett. 85 (000) 39. [] M. Christandl, N. Datta, A. Ekert, et al., Phys. Rev. Lett. 9 (00) [3] F. Vestraete, M.A. Martin-Delgado, and J.I. Cirac, Phys. Rev. Lett. 9 (00) [] I.A. Grigorenko and D.V. Khveshchenko, Phys. Rev. Lett. 95 (005) [5] J.P. Barjaktarevic, R.H. McKenzie, J. Links, et al., Phys. Rev. Lett. 95 (005) [6] I. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki, Commun. Math. Phys. 115 (1988) 77. [7] A. Klümper, A. Schadschneider, and J. Zittartz, J. Phys. A (1991) L955; Z. Phys. B 87 (199) 81. [8] M. Fannes, B. Nachtergaele, and R.F. Werner, Commun. Math. Phys. 1 (199) 3. [9] F. Verstraete, D. Porras, and J.I. Cirac, Phys. Rev. Lett. 93 (00) 705. [10] F. Verstraete, J.J. Garcia-Ripoll, and J.I. Cirac, Phys. Rev. Lett. 93 (00) 070. [11] F. Verstraete and J.I. Cirac, cond-mat/050510; T.J. Osborne, quant-ph/ ; M.B. Hastings, cond-mat/ [1] M. Zwolak and G. Vidal, Phys. Rev. Lett. 93 (00) 0705; G. Vidal, Phys. Rev. Lett. 93 (00) [13] D.P. Garcia, F. Verstraete, M.M. Wolf, and J.I. Cirac, Quantum Inf. Comput. 7 (007) 01. [1] J.M. Zhu, Chin. Phys. Lett. 5 (008) 357. [15] J.M. Zhu, Commun. Theor. Phys. 5 (010) 373. [16] Zhang Guo-Feng, Physics (013) 55. [17] Ma Xiao-San, Qiao Ying, et al., Sci. China Phys. Mech. Astronom. 56 (013) 600. [18] Zhou Tao, Long Gui-Lu, et al., Physics (013) 5. [19] Wen Wei, Sci. China Phys. Mech. Astronom. 56 (013) 97. [0] Zhao Hui, Zhang Xing-Hua, Fei Shao-Ming, et al., Chin. Sci. Bull. 58 (013) 33. [1] Man Zhong-Xiao, Su Fang, and Xia Yun-Jie, Chin. Sci. Bull. 58 (013) 3. [] Cao Wan-Cang, Liu Dan, Pan Feng, et al., Sci. Chin. Sci. Bull. Sci. China Ser. G-Phys. Mach. Astronom. 9 (006) 606. [3] D. Liu, X. Zhao, and G.L. Long, Commun. Theor. Phys. 5 (010) 85. [] D. Liu, X. Zhao, and G.L. Long, Commun. Theor. Phys. 9 (008) 39. [5] H. Wu, X. Zhao, and Y.S. Li, Int. J. Quantum Inform. 8 (010) 1169 [6] S. Muralidharan and P.K. Panigrahi, Phys. Rev. A 77 (008) [7] S. Muralidharan and P.K. Panigrahi, Phys. Rev. A 78 (008) [8] J.M. Zhu, Chin. Phys. C 36 (01) 311. [9] Cao Ye, Li Hui, and Long Gui-Lu, Chin. Sci. Bull. 58 (01) 8. [30] B. Dakié, V. Vedral, and C. Brukner, Phys. Rev. Lett. 105 (010)