Local quantum Fisher information and one-way quantum deficit in spin- 1 2 XX Heisenberg chain with three-spin interaction
LLocal quantum Fisher information and one-way quantum deficitin spin- X X
Heisenberg chain with three-spin interaction
Biao-Liang Ye, ∗ Bo Li,
2, 1
Xiao-Bin Liang,
2, 1 and Shao-Ming Fei
3, 4, † Quantum Information Research Center,Shangrao Normal University, Shangrao 334001, China School of Mathematics & Computer Science,Shangrao Normal University, Shangrao 334001, China School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany (Dated: August 31, 2020)We explore quantum phase transitions in the spin-1/2 XX chain with three-spininteraction in terms of local quantum Fisher information and one-way quantumdeficit, together with the demonstration of quantum fluctuations. Analytical resultsare derived and analyzed in detail. I. INTRODUCTION
Quantum entanglement plays a vital role in quantum information processing [1]. Asan important resource, quantum entangled states have been used in quantum teleportation[2], remote state preparation [3], secure quantum-communications network [4], etc. Be-sides quantum entanglement, quantum discord characterizes non-classical correlations [5].The one-way quantum deficit [6] is another key measure to describe quantum correlation[7]. While the quantum Fisher information [8, 9] is important in the estimation accuracyscenarios.On the other hand, the quantum phase transitions have received much attention in con-densed matter physics [10]. The quantum fluctuations are able to be illustrated by quantumcorrelations. In Ref. [11] the role of entanglement played in phase transition and theory of ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ qu a n t - ph ] A ug critical phenomena in XY system has been investigated. The quantum discord and entan-glement between the nearest-neighbor qubits in an infinite (spin-1/2) chain described by theHeisenberg model (XXZ Hamiltonian) were investigated in [12], and the critical points asso-ciated with quantum phase transitions at finite temperature had been analyzed. In Ref.[13],the authors studied the effects of Dzyaloshinskii-Moriya interaction on pairwise quantumdiscord, entanglement, and classical correlation in the anisotropic XY spin-half chain. Ithas been shown that the quantum discord can be useful to highlight the quantum phasetransition, especially for the long-distance spins, while entanglement decays rapidly. Thequantum discord has been also used to show the quantum phase transition in XX modelin [14]. In Ref. [15] the authors connected the local quantum coherence based on Wigner-Yanase skew information and the quantum phase transitions. The local quantum coherenceand its derivatives are used effectively in detecting different types of quantum phase transi-tions in different spin systems. In Ref. [16] the authors introduced a coherence susceptibilityapproach in identifying quantum phase transitions induced by quantum fluctuations.Although different measures of quantum correlations have been used to characterize quan-tum phase transitions in different spin chain systems, both local quantum Fisher informationand one-way quantum deficit have not been adopted to study quantum phase transition inHersenberg XX models. In this paper, we investigate the quantum phase transitions ofthe XX chain with three spin interaction. We explore the quantum fluctuation via bothlocal quantum Fisher information and one-way quantum deficit to investigate the quantumphase transition. We review the basic definitions of local quantum Fisher information andone-way quantum deficit in Sec. II. In Sec.III, the Hersenberg spin- is introduced and themain results are presented. Conclusions are given in Sec. IV. II. PRELIMINARIES
We first recall the basic definitions of local quantum Fisher information and one-wayquantum deficit.
Local quantum Fisher information
Quantum Fisher information (QFI) is recognized as themost widely used quantity for characterizing the ultimate accuracy in parameter estimationscenarios. Recently, many efforts have been made toward evaluating the dynamics of QFIto establish the relevance of quantum entanglement in quantum metrology. It has beendemonstrated that quantum entanglement leads to a notable improvement of the accuracyof parameter estimation. It is natural to ask whether quantum correlations beyond quantumentanglement can be related to the precision in quantum metrology protocols.Generally, for an arbitrary quantum state ρ θ that depends on a variate θ , the QFI isexpressed by [17], F ( ρ θ ) = 14 Tr[ ρ θ L θ ] . (1)Here the symmetric logarithmic derivative L θ is get as the solution of the following equation ∂ρ θ ∂θ = 12 ( L θ ρ θ + ρ θ L θ ) . (2)The parametric states ρ θ can be derived from an initial probe state ρ subjected to a unitarytransformation U θ = e − iHθ which is dependent on θ and a Hermitian operator H , i.e., ρ θ = U † θ ρU θ . Thus F ( ρ θ ) is given by F ( ρ, H ) = 12 (cid:88) i (cid:54) = j ( p i − p j ) p i + p j |(cid:104) ψ i | H | ψ j (cid:105)| , (3)where ρ = (cid:80) i =1 p i | ψ i (cid:105)(cid:104) ψ i | is spectral decomposition of ρ , p i ≥ (cid:80) i =1 p i = 1.Now consider a bipartite quantum state ρ AB in the Hilbert space W = W A ⊗ W B . Weassume that the dynamics of the first subsystem is subjected by the local phase shift trans-formation e − iθH A , with H A = H a ⊗ I B the local Hamiltonian. Therefore QFI reduces to localquantum Fisher information (LQFI), F ( ρ, H A ) = Tr( ρH A ) − (cid:88) i (cid:54) = j p i p j p i + p j |(cid:104) ψ i | H A | ψ j (cid:105)| . (4)Local quantum Fisher information was introduced to deal with pairwise quantum mea-sures of discord type. The local quantum Fisher information Q ( ρ ) is defined as the minimumquantum Fisher information over all local Hamiltonians H A acting on the subsystem A [18], Q ( ρ ) = min H A F ( ρ, H A ) . (5)For local Hamiltonian H a = (cid:126)σ · (cid:126)r , with | (cid:126)r | = 1 and (cid:126)σ = ( σ x , σ y , σ z ) the Pauli matrices, onehas Tr( ρH A ) = 1, and the second term in (4) can be written as (cid:88) i (cid:54) = j p i p j p i + p j | ψ i | H A | ψ j (cid:105)| = (cid:88) i (cid:54) = j (cid:88) l,k =1 p i p j p i + p j (cid:104) ψ i | σ l ⊗ I B | ψ j (cid:105)(cid:104) ψ j | σ k ⊗ I B | ψ i (cid:105) = (cid:126)r † · T · (cid:126)r, (6)where the elements of the 3 × T are given by T lk = (cid:88) i (cid:54) = j p i p j p i + p j (cid:104) ψ i | σ l ⊗ I B | ψ j (cid:105)(cid:104) ψ j | σ k ⊗ I B | ψ i (cid:105) . (7)To minimize F ( ρ, H A ), it is necessary to maximize the quantity (cid:126)r † · T · (cid:126)r over all unitvectors (cid:126)r . The maximum value coincides with the maximum eigenvalue of T . Hence, theminimal value of local quantum Fisher information Q ( ρ ) is Q ( ρ ) = 1 − λ max ( T ) , (8)where λ max denotes the maximal eigenvalue of the symmetric matrix T defined by (7). One-way quantum deficit (OWQD)
The one-way quantum deficit is defined as the differ-ence of the von Neumann entropy of a bipartite state, ρ AB , before and after a measurementperformed on, without a loss of generality, subsystem B [7],∆ = min Π iA S ( (cid:88) i Π Bk ( ρ AB )) − S ( ρ AB ) , (9)where Π Bk is the measurement on subsystem B and S ( ρ AB ) = − Tr ρ AB log ρ AB is the vonNeumann entropy. Throughout the article, log is in base 2. The minimum is taken over alllocal measurements Π Bk . The states after the projective measurement can be expressed as (cid:101) ρ AB = (cid:88) k ( I ⊗ Π k ) ρ AB ( I ⊗ Π k ) † . (10)The post-measurement states is given by ρ kAB = 1 p k ( I ⊗ Π k ) ρ AB ( I ⊗ Π k ) † , (11)where p k = Tr[( I ⊗ Π k ) ρ AB ( I ⊗ Π k ) † ] . (12)Here Π k ( k = 0 ,
1) are the general orthogonal projectorsΠ k = V | k (cid:105)(cid:104) k | V † , (13)where V belongs to the special unitary group SU (2). The rotations V may be parametrizedby two parameters θ and φ , respectively, V = cos( θ/ − e − iφ sin( θ/ e iφ sin( θ/
2) cos( θ/ , (14)with θ ∈ [0 , π ] and φ ∈ [0 , π ]. III. QUANTUM PHASE TRANSITION OF THE HEISENBERG- XX SPINCHAIN MODEL WITH THREE SPIN INTERACTION
The Hamiltonian of the Heisenberg spin- XX chain can be given as [19], H = N (cid:88) l =1 − J ( S xl S xl +1 + S yl S yl +1 + ∆ S zl S zl +1 ) − J (cid:48) { ( S xl − S zl S yl +1 − S yl − S zl S xl +1 )+ ∆( S yl − S xl S zl +1 − S zl − S xl S yl +1 ) + ∆( S zl − S yl S xl +1 − S xl − S yl S zl +1 ) } . (15)Here N is the total number of spins, S ql ( q = x, y, z ) are spin operators of S = 1 / l , J is the nearest-neighbor Heisenberg exchange coupling, J (cid:48) is the strength ofthree spin interaction, and ∆ denotes the anisotropy parameter. The model shows severalquantum phases depending on the parameters ∆ and J (cid:48) /J . The same Hamiltonian is usedto investigate the current-carrying states for the system with only the nearest-neighborinteractions, where the three-spin terms play the role of the Lagrange multiplier.If ∆ = 0, the Hamiltonian (15) reduces to a free spinless fermion model, H = N (cid:88) l =1 − J ( S xl S xl +1 + S yl S yl +1 ) − J (cid:48) { ( S xl − S zl S yl +1 − S yl − S zl S xl +1 ) } . (16)Applying the Jordan-Wigner transformation, S xl = 1 / l − n =1 (1 − c † n c n )( c † l + c l ); S yl = 1 / i Π l − n =1 (1 − c † n c n )( c † l − c l ); S zl = c † l c l − / , (17) H can be rewritten as H = N (cid:88) l =1 [ − J/ c † l c l +1 + h.c. ) + J (cid:48) / i ( c † l c l +2 − h.c. )] , (18)which can be diagonalized by means of the Fourier transformation, H = (cid:88) k (cid:15) ( k ) c † k c k , (19)where the energy dispersion (cid:15) ( k ) = − J [cos k − α/ k )] , (20)with α = J (cid:48) /J .The matrix form of the Hamiltonian form can be denoted as ρ = u ij ω ij y ij y ij ω ij
00 0 0 u ij , (21)where all the elements of the matrix can be written in terms of spin-spin correlation functions, u ij = 14 + (cid:104) S zi S zj (cid:105) ,ω ij = 14 − (cid:104) S zi S zj (cid:105) ,y ij = (cid:104) S xi S xj (cid:105) + (cid:104) S yi S yj (cid:105) , with (cid:104) S qi S qj (cid:105) ( q = x, y, z ) the two-point spin-spin correlation functions at sites i and j , andthe expectation value taken over all the quantum states.Using the method proposed by Lieb, Schultz, and Mattis [20], one can also calculate thespin-spin correlation functions, (cid:104) S xl S xl + m (cid:105) = (cid:104) S yl S yl + m (cid:105) = 14 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G l,l +1 G l,l +2 · · · G l,l + m G l,l G l,l +1 · · · G l,l + m − · · · · · · . . . · · · G l,l − m +2 G l,l − m +3 · · · G l,l +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (22)and (cid:104) S zl S zl + m (cid:105) = − / G l,l + m ) , (23)with G l,l + m = mπ sin( m π ) α < , mπ [1 − ( − m ] sin( m arcsin(1 /α )) α ≥ . (24)Let t = 4 (cid:104) S xl S xl + m (cid:105) , t = 4 (cid:104) S yl S yl + m (cid:105) and t = 4 (cid:104) S zl S zl + m (cid:105) . The 3 × T is of theform, T = ( t +1) ( t + t − ) t − ( t +1) ( t + t − ) t −
00 0 t + t − t − , (25) α - - - - - α LQFIOWQD dLQFI / d α dOWQD / d α ( a ) ( b ) FIG. 1: m = 1: (a) LQFI and OWQD with respect to α . The orange dashed line denotes LQFI.The orange solid line shows OWQD. (b) The derivatives of LQFI (dashed orange line) and OWQD(solid orange line) with respect to α , respectively. with eigenvalues (cid:40) t + t − t − , ( t + 1) (2 t + t − t − t + 1) , ( t + 1) (2 t + t − t − t + 1) (cid:41) . (26)From (8) we have the LQFI, LQF I = 1 − max[ ( t + 1) (2 t + t − t − , t + t − t − . (27)By tedious calculation, we can also work out the OWQD. The analytical expression ofOWDQ for the Heisenberg XX spin chain is given by∆ = − t ) log (1 + 2 t ) − − t ) log (1 − t )+ 14 [(1 − t + 2 t ) log(1 − t + 2 t )+(1 − t − t ) log(1 − t − t )+2(1 + t ) log(1 + t )] , (28)with the optimal value attained at φ = 0 and θ = π/ XX spin chain system.Fig. 1 shows the LQFI, OWQD and their derivatives with respect to α . In Fig. 1(a), thedashed line denotes LQFI, while the solid line stands for OWQD. One sees that in region[0 , α - - - - - α LQFIOWQD dLQFI / d α dOWQD / d α ( a ) ( b ) FIG. 2: m = 2: (a) LQFI and OWQD with respect to α . The orange dashed (solid) line denotesLQFI (OWQD). (b) Dashed (solid) orange line denotes the derivative of LQFI (OWQD) withrespect to α . then slowly in the region α >
1. They have the same trends. However, LQFI is greater thanOWQD. They approach zero when α goes to infinity.The derivatives of LQFI and OWQD with respect to α are shown in Fig. 1(b), from whichwe see that the quantum phase transition happens at α = 1. In the region α ∈ [0 , α >
1, the derivatives of OWQD is greaterthan that of LQFI, meaning that the slope related to LQFI is steeper than to OWQD.Fig.2 shows LQFI and OWQD, and their derivatives with respect to α when m = 2.The LQFI and OWQD for m = 2 are smaller than that for m = 1 in Fig.2(a), respectively.Around the region α ∈ [1 , m = 1.When α gets larger, the LQFI is approximately coincident with the OWQD. Both quantumcorrelation measures LQFI and OWQD show quantum phase transition in Fig. 2(b).Fig. 3 shows the behavior of LQFI vs α (dashed blue line). Fig. 4 shows the behaviorof OWQD vs α (solid blue line). The insets show the quantum phase transition relatedto their derivatives. One can see that when m increases, the slopes of the lines get larger.However, both of them show quantum phase transition of the Heisenberg XX model by thefirst derivatives at α = 1. α L Q F I - - α d L Q F I / d α FIG. 3: LQFI and its derivative (inset) with respect to α for m = 3. α O W Q D - - - - - α d O W Q D / d α FIG. 4: OWQD and its derivative (inset) with respect to α for m = 3. The inset shows thequantum phase transition related to its derivatives. IV. CONCLUSIONS
We have studied the quantum phase transitions in Heisenberg spin- XX spin model,showing that the quantum phase transition happens at α = 1. Both quantum measures, localquantum Fihser information and one-way quantum deficit, are able to show the quantumfluctuation and the quantum phase transition for the Heisenberg spin- XX spin system.Our results may highlight the corresponding experimental demonstrations of the quantumfluctuation and the quantum phase transition in the Heisenberg spin- XX spin systems.0 Acknowledgment
This work was supported by the Key-Area Research and Development Program ofGuangdong Province (Grant No. 2018B030326001), the NKRDP of China (GrantNo. 2016YFA0301802), the National Natural Science Foundation of China (Grant Nos.11675113, 11765016, 11847108, and 11905131), Beijing Municipal Commission of Educa-tion (Grant No. KZ201810028042), the Beijing Natural Science Foundation (Z190005),the Natural Science Foundation of Jiangxi Province (Grant Nos. 20192BAB212005,20192ACBL20051), and Jiangxi Education Department Fund (Grant No. KJLD14088 andGJJ190888). [1] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev.Mod. Phys. , 865 (2009).[2] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleportingan unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys.Rev. Lett. , 1895 (1993).[3] C. H. Bennett, D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and W. K. Wootters,Remote state preparation, Phys. Rev. Lett. , 077902 (2001).[4] J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai,G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J.Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A.Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, Satellite-based entanglementdistribution over 1200 kilometers, Science , 1140 (2017) .[5] K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, The classical-quantum boundaryfor correlations: Discord and related measures, Rev. Mod. Phys. , 1655 (2012).[6] B.-L. Ye, Y.-K. Wang, and S.-M. Fei, One-way quantum deficit and decoherence for two-qubitX states, Int. J. Theor. Phys. , 2237 (2016).[7] A. Streltsov, H. Kampermann, and D. Bruß, Linking quantum discord to entanglement in ameasurement, Phys. Rev. Lett. , 160401 (2011).[8] D. Petz, Covariance and Fisher information in quantum mechanics, Journal of Physics A: Mathematical and General , 929 (2002).[9] B.-L. Ye, B. Li, Z.-X. Wang, X. Li-Jost, and S.-M. Fei, Quantum Fisher information andcoherence in one-dimensional XY spin models with Dzyaloshinsky-Moriya interactions, Sci.China-Phys. Mech. Astron. , 110312 (2018).[10] S. Sachdev, Quantum Phase Transitions (Cambridge University Press , Cambridge, 1999).[11] A. Osterloh, L. Amico, G. Falci, and R. Fazio, Scaling of entanglement close to a quantumphase transition, Nature , 608 (2002).[12] T. Werlang, C. Trippe, G. A. P. Ribeiro, and G. Rigolin, Quantum correlations in spin chainsat finite temperatures and quantum phase transitions, Phys. Rev. Lett. , 095702 (2010).[13] B.-Q. Liu, B. Shao, J.-G. Li, J. Zou, and L.-A. Wu, Quantum and classical correlations in theone-dimensional XY model with dzyaloshinskii-moriya interaction, Phys. Rev. A , 052112(2011).[14] B.-Q. Liu, B. Shao, and J. Zou, Quantum and classical correlations in isotropic XY chain withthree-site interaction, Commun. Theor. Phys. , 46 (2011).[15] Y.-C. Li and H.-Q. Lin, Quantum coherence and quantum phase transitions, Sci. Rep. ,26365 (2016).[16] J.-J. Chen, J. Cui, Y.-R. Zhang, and H. Fan, Coherence susceptibility as a probe of quantumphase transitions, Phys. Rev. A , 022112 (2016).[17] A. Slaoui, L. Bakmou, M. Daoud, and R. A. Laamara, A comparative study of local quantumFisher information and local quantum uncertainty in Heisenberg XY model, Phys. Lett. A , 2241 (2019).[18] S. Kim, L. Li, A. Kumar, and J. Wu, Characterizing nonclassical correlations via local quantumFisher information, Phys. Rev. A , 032326 (2018).[19] P. Lou, W.-C. Wu, and M.-C. Chang, Quantum phase transition in spin- XX Heisenbergchain with three-spin interaction, Phys. Rev. B , 064405 (2004).[20] E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann.Phys.16