Majorization relation in quantum critical systems
Lin-Ping Huai, Yu-Ran Zhang, Si-Yuan Liu, Wen-Li Yang, Shi-Xian Qu, Heng Fan
aa r X i v : . [ qu a n t - ph ] M a y Majorization relation in quantum critical systems
HUAI Lin-Ping,
ZHANG Yu-Ran, ∗ LIU Si-Yuan,
3, 2
YANG Wen-Li, QU Shi-Xian, † and FAN Heng
2, 4, ‡ Institute of Theoretical & Computational Physics, School of Physics andInformation Technology, Shaanxi Normal University, Xi’an 710062, China Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Institute of Modern Physics, Northwest University, Xi’an 710069, China Collaborative Innovation Center of Quantum Matter, Beijing , China (Dated: September 10, 2018)The most basic local conversion is local operations and classical communications (LOCC), which is also themost natural restriction in quantum information processing. We investigate the conversions between the groundstates in quantum critical systems via LOCC and propose an novel method to reveal the different convertibilityvia majorization relation when a quantum phase transition occurs. The ground-state local convertibility in theone-dimensional transverse field Ising model is studied. It is shown that the LOCC convertibility changesnearly at the phase transition point. The relation between the order of quantum phase transitions and the LOCCconvertibility is discussed. Our results are compared with the corresponding results using the R´enyi entropy andthe LOCC convertibility with assisted entanglement.
PACS numbers: 64.60.A-;03.67.Mn;05.70.Jk
Many developments in quantum-information processing(QIP) [1] unveiling the rich structure of quantum states and thenature of entanglement have offered many insights into quan-tum many-body physics [2]. Concepts from QIP have gener-ated many alternative indictors of quantum phase transitions,which has become a focus of attention in detecting a numberof critical points. For example, entanglements measured byconcurrence [3], negativity [4], geometric entanglement [5],and von Neumann entropy [6, 7] have been investigated inseveral critical systems. From another viewpoint of quantumcorrelations, other concepts in quantum information includ-ing mutual information [8], quantum discord [9] and globalquantum discord [10], have also been used for detecting quan-tum phase transitions. Other investigations in entanglementspectra [11–13] and fidelity [14] as well as the fidelity sus-ceptibility of the ground state show their abilities in exploringnumerous phase transition points in various critical systems,as well. These observations have achieved great success inunderstanding the deep nature of the different phase transi-tions with the tools of quantum information science harnessedin the analysis of quantum many-body systems. The reverse,however, seems unclear and tough.From a new point of view, Jian Cui and his collaboratorsreveal that the systems undergoing quantum phase transitionwill also exhibit different operational properties from the per-spective of QIP [15, 16]. They demonstrate with several mod-els that nearly at the critical points, the entanglement-assistedlocal operations and classical communications (ELOCC) con-vertibility decided via the R´enyi entropy interception changes ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] suddenly, see Fig. 1. The R´enyi entropy is defined as [17] S α ( ρ A ) = 11 − α log Tr ρ αA (1)and it tends to the von Neumann entropy as α → . The un-expected results suggest that not only are the tools of quantuminformation useful as alternative signatures of quantum phasetransitions, but the study of quantum phase transitions mayalso offer additional insight into QIP. However, the most basicand natural local conversion is local operations and classicalcommunications (LOCC), which has fabulous applications inQIP, such as teleportation [18], quantum states discrimination[19], and quantum channel corrections [20].In this paper, we investigate the conversions between theground states in quantum critical systems via LOCC, and pro-pose an novel method to reveal the different convertibility viamajorization relation [21] when a quantum phase transitionoccurs. We study the one-dimensional transverse field Isingmodel and show that the LOCC convertibility changes nearlyat the phase transition point. Presenting the relation betweenthe order of quantum phase transitions and the LOCC convert-ibility, we compare our results with the corresponding results α S α h=0.5h=0.6h=0.7h=0.8h=0.9h=1.0 (a) α S α h=1h=1.1h=1.2h=1.3h=1.4h=1.5 (b) FIG. 1: (Color online) R´enyi entropy for the ground state of the trans-verse field Ising model v.s. α . The R´enyi entropies are intercepted as(a) h ≤ , while they are nonintercepted as (b) h ≥ , which gives aphysical significance to QIP from quantum. transitions h f ( h ) N=4N=6N=8N=10N=12N=14N=16N=18N=20N=22 (a) h f (h)f (h) (b) −4−2024 x 10 −4 h −4 h h (II)(II) (I) (I) N: 22 <- 4N: 4 -> 22 f’ (h)f’ (h) (c) FIG. 2: (Color online) (a) Functions f ( h ) against field parameter h for Ising rings containing from to spins at zero temperature. Forclearness, we merely perform the cases for even spin numbers. It isclear that f ( h ) is monotonically increasing for all system sizes. (b)We show f ( h ) (dashed red lines) and f ( h ) (solid blue lines) fordifferent system sizes ranging from to . Both f ( h ) and f ( h ) firstly decrease and then increase. (c) Derivatives f ′ ( h ) (dashed redlines) and f ′ ( h ) (solid blue lines) against h for N ranging from to . The minimum points for f ( h ) and f ( h ) appear where f ′ ( h ) =0 and f ′ ( h ) = 0 . using the ELOCC convertibility.The majorization relations provide a necessary and suf-ficient condition for the LOCC convertibility. For two bi-partite pure states | ψ i = P dk =1 q λ kψ | kk i AB and | ϕ i = P dk =1 q λ kϕ | kk i , if and only if the majorization relations l X k =1 λ kψ ≥ l X k =1 λ kϕ (2)are satisfied for all ≤ l ≤ d (expressed as λ ψ ≻ λ ϕ ),state | ϕ i can be transformed with 100% probability of suc-cess to | ψ i by LOCC. Otherwise these two states are incom-parable, i.e., | ψ i cannot be converted to | ϕ i by LOCC, andvice versa. For example, | ψ i = √ . | i + √ . | i + √ . | i , | ψ i = √ . | i + √ . | i + √ . | i and | ψ i = √ . | i + √ . | i + √ . | i . It can be eas-ily checked that λ ψ ≺ λ ψ which leads to that | ψ i can beconverted to | ψ i with certainty. For states | ψ i and | ψ i ,no majorization relations can be fulfilled (i.e. λ ψ ⊀ λ ψ and λ ψ ⊁ λ ψ ) and it indicates that neither state can be convertedto the other with certainty.Next, given the transverse field Ising model in the zero-temperature case, we use majorization relations to investigatethe relationship between the critical points and the LOCC con-vertibility of the ground state. The Hamiltonian for a chain of h c h Majoriza(cid:127)on rela(cid:127)on λ h+ ∆ λ h |G(h) ∆ |G(h+ ) Phase 1 Phase 2
LOCC
Conver(cid:127)bilityNo majoriza(cid:127)on rela(cid:127)ons λ h+ ∆ λ h |G(h+ ) ∆ |G(h) LOCC
No conver(cid:127)bility λ h+ ∆ λ h |G(h+ ) ∆ |G(h) LOCC or FIG. 3: (Color online) Different LOCC convertibilities for twophases in the transverse field Ising model. N spin- particles reads H = − N X i =1 (cid:0) σ xi σ xi +1 + hσ zi (cid:1) , (3)where periodic boundary conditions are assumed: N +1 → . σ xi and σ zi stand for the Pauli matrices, h is the field parameter.A second-order quantum phase transition takes place at h = 1 .For h > , it is the paramagnetic phase, while it is the ferro-magnetic phase for < h < . We study the ground states ofthis model for system sizes N = 4 , · · · , with the field pa-rameter h varying from . to . . The ground states labelledas | G ( h ) i AB are obtained by exactly diagonalizing the wholeHamiltonian (3). This new proposal is also worth further in-vestigation by other numerical methods such as Lanczos al-gorithm and density matrix renormalization group (DMRG),which are not included in this paper.We consider three largest eigenvalues of the reduced statesof any two neighbor spins as λ ( h ) , λ ( h ) , and λ ( h ) in de-scending order. In order to detect the majorization relationsbetween two ground states | G ( h ) i AB and | G ( h + ∆) i AB given some infinitesimal ∆ , we should judge the monotonici-ties of three functions for the field parameter h : f ( h ) ≡ λ ( h ) , (4) f ( h ) ≡ λ ( h ) + λ ( h ) , (5) f ( h ) ≡ λ ( h ) + λ ( h ) + λ ( h ) . (6)Therefore, three cases will be met: ( i ) when monotonicitiesof these three functions are all non-increasing, | G ( h + ∆) i AB can be converted to | G ( h ) i AB by LOCC with certainty; ( ii ) when monotonicities of these three functions are all non-decreasing | G ( h ) i AB can be converted to | G ( h + ∆) i AB byLOCC with certainty; ( iii ) except for these two cases, neitherstate can be converted to the other via LOCC with certainty.Function f ( h ) against the field parameter for system size N varying from to is shown in Fig. 2(a), where wefind f ( h ) is monotonically increasing for h ∈ [0 . , . .In Fig. 2(b), for system sizes ranging from to , dashedred lines and solid blue lines denote to f ( h ) and f ( h ) ver-sus h , respectively. We also present the derivatives of f ( h ) and f ( h ) in Fig. 2(c). Both functions firstly decrease andthen increase starting nearly at the points h c ≃ , where f ′ ( h ) f ′ ( h ) = 0 , f ′ ( h ) ≥ and f ′ ( h ) ≥ are fulfilled. Thus, (a) (b) FIG. 4: Finite-size scaling analysis of the minimum points of f ( h ) and f ( h ) in Ising model: (a) is for h min and (b) is for h min . we can conclude for the transverse field Ising model that themajorization relations λ h +∆ ≻ λ h hold when h > h c , andmeanwhile, | G ( h ) i AB can be transformed to | G ( h + ∆) i AB by LOCC with certainty. For the rest range of h , no majoriza-tion relation can be found. These results indicate that there isa distinct change in the properties of ground states at the criti-cal points h c , see Fig. 3. It seems credible to detect the phasetransition points via the majorization relations for the LOCCconvertibility, and in contrary, it suggests that the LOCC con-vertibilities of the ground states of transverse field Ising modelin two phases are different.To get rid of the finite-size effect, we give a scaling analysisof the critical points h c in Fig. 4. The parameters h min and h min for the minimum points of f ( h ) and f ( h ) for differentsystem sizes are plotted in Fig. 4(a) and 4(b), respectively.They can be fitted as h min = 1 . /N . + 1 . withaccuracy . × − and h min = − . /N . + 1 . with accuracy . × − . Therefore, the critical point forlarge N should be chosen as h c = 1 . on the basis ofthe majorization relations and the LOCC convertibility. As N → ∞ , it can clearly demonstrate that when h ≥ . , | G ( h ) i AB can be transformed to | G ( h + ∆) i AB by LOCCwith certainty, and in the region < h < . , neither statecan be converted to the other via LOCC with certainty.Then we compare our results with the results obtained bythe ELOCC convertibility with R´enyi entropy. States | ψ i AB can be transformed to | ψ ′ i AB via ELOCC if and only if theirR´enyi entropies satisfy S α ( ρ A ) ≥ S α ( ρ ′ A ) for all α given ρ A and ρ ′ A the reduced density matrices of | ψ i AB and | ψ ′ i AB ,respectively [22]. In Ref. [15], it shows that the phase transi-tion point of the transverse field Ising model via the ELOCCproposal with R´enyi entropy is h cE = 0 . as N → ∞ which offers a higher accuracy for detecting the critical pointsfor the transverse field Ising model. It demonstrates thatthe point where the ELOCC convertibility suddenly changesstands closer to the critical points than the one at which theLOCC convertibility changes. Combined with this conclu-sion, a clearer and further description of the local convert-ibility of the ground states of the transverse field Ising modelcan be presented: In the region < h < . within theferromagnetic phase, neither LOCC nor ELOCC convertibil-ities exist, in the region h > . within the paramagneticphase, both LOCC and ELOCC convertibilities exist, and inthe small interval . < h < . around the criticalpoint h = 1 , merely the ELOCC convertibility exists.In conclusion, we investigate the majorization relations andthe LOCC convertibility in quantum critical systems. We de-velop a proposal to describe the LOCC conversion propertiesof quantum critical systems by examining the majorizationrelations. We apply this proposal to study one-dimensionaltransverse field Ising model, which shows that the LOCC con-vertibility changes at h c = 1 . nearly at the critical point.ELOCC convertibility and LOCC convertibility in most ar-eas of the paramagnetic phase are both stronger than those inthe ferromagnetic phase, however, in small intervals aroundthe critical ponit h = 1 , merely the ELOCC convertibilityexists. In the ferromagnetic phase where < h < , theground states can nearly convert to each other by LOCC orELOCC; whereas in the paramagnetic phase where h > , thesituation is slightly more complicated. The LOCC convert-ibility via majorization relations applied for quantum phasetransition is a new method. It will help us understand from adifferent view point the complicated phenomena of quantumcritical systems. This LOCC proposal in detecting quantumphase transition can play a complementary role to the ELOCCmethod; it should be applicable in other systems similar as inthe one-dimensional transverse field Ising model. This pa-per will enlighten extensive studies of quantum phase transi-tions from the perspective of local convertibility, e.g., finite-temperature phase transitions and other quantum many-bodymodels. Acknowledgments
We thank Jun-Peng Cao, Dong Wang and Yu Zeng forvaluable discussions. This work is supported by NSFC(11175248), 973 program (2010CB922904). [1] Bennett C H 1995
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