Locality of three-qubit Greenberger-Horne-Zeilinger-symmetric states
aa r X i v : . [ qu a n t - ph ] F e b Locality of three-qubit Greenberger-Horne-Zeilinger-symmetric states
Dian Zhu, Gang-Gang He, and Fu-Lin Zhang ∗ Department of Physics, School of Science, Tianjin University, Tianjin 300072, China (Dated: February 22, 2021)The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamentalproblems in quantum physics. Existing studies on this topic to date were limited to the entangle-ment classification according to the numbers of particles enrolled. Equivalence under stochastic localoperations and classical communication provides a more detailed classification, e. g. the genuinethree-qubit entanglement being divided into W and GHZ classes. We construct two families of localmodels for the three-qubit Greenberger-Horne-Zeilinger (GHZ)-symmetric states, whose entangle-ment classes have a complete description. The key technology of construction the local models inthis work is the GHZ symmetrization on tripartite extensions of the optimal local-hidden-state mod-els for Bell diagonal states. Our models show that entanglement and nonlocality are inequivalentfor all the entanglement classes (biseparable, W, and GHZ) in three-qubit systems.
I. INTRODUCTION
Several concepts of nonclassical correlations in com-posite quantum systems have been presented to revealsignificant differences between the quantum and classi-cal worlds [1–5]. Many of them can be traced back tothe early days of quantum mechanics, and play key rolesin different quantum information processes. In genereal,these correlations arise from coherent superposition ofcomposite quantum systems, and are equivalent to eachother in pure states. The classical probabilities in mixedstates divide them into hierarchies, and thereby lead todifferent classical-quantum boundaries[5]. The hierarchyof quantum correlations is an important issue in bothfundamental quantum theory and practical applicationsof quantum information.Studies on the hierarchy of quantum correlations oftenneed carefully examining their related classicality. En-tanglement exists in nonseparable states, while a separa-ble state is defined as the one can be expressed as a mix-ture of product states [2]. Nonlocality [3, 4] is demon-strated by the absence of local-hidden-variable (LHV)models for the outcomes of local measurements, whichcan be detected by the violation of Bell inequality. Thenonlocal states are a strict subset of the entangled states,as all separable states and a part of entangled states canbe modeled by LHV theories. The existence of LHV mod-els for separable states is trivial, but for entangled statesis an important progress, which is found by Werner [6]in a family of bipartite mixed states, nowadays known asthe Werner states.Since Werner putted forward his original results [6], therelation between entanglement and nonlocality for bipar-tite quantum systems has been intensively discussed inmany directions, including performing general measure-ments nonsequentially [7] or sequentially [8] and in theschemes using several copies [9–13]. However very fewworks for multipartite systems have been reported. Here, ∗ Corresponding author: fl[email protected] the interesting also the difficult point is that multipartitestates offer a richer variety of different types of entangle-ment and nonlocality. The notions of genuine multipar-tite entanglement (GME) and genuine multipartite non-locality (GMNL) among a whole system have been pro-posed to distinguish them with the weak entanglement insubsystems. Research on the locality of multipartite en-tangled states was initiated by T´oth and Ac´ın [14], whofound a fully local model for a family of three-qubit stateswith GME. Recent two works extended the investigationto the systems with any number of parties. Augusiak et al. [15] showed GME and GMNL are inequivalent byconstructing a bilocal model, in which the parties are sep-arated into two groups, for a class of GME states. Bowles et al. [16] found that there exist states with GME ad-mitting a fully LHV model, which can never lead to anyBell inequality violation for general nonsequential mea-surements.On the other hand, Einstein-Podolsky-Rosen (EPR)steering, the quantum correlation lying between nonlo-cality and entanglement, has been widely studied boththeoretically and experimentally since its operational def-inition was provided by Wiseman et al. [17]. It exists inthe entangled states whose unnormalized postmeasuredstates, after one-side local measurements, cannot be de-scribed by a local-hidden-state (LHS) model. A LHSmodel is a particular case of a LHV model with the hid-den variable being a local state. The EPR steering isnot only an important resource in quantum informationprocesses [18–20], but also acts as a powerful tool in re-searches of entanglement and nonlocality, such as in theconstruction of counterexamples to the Peres conjecture[21–23], generalizing Gisin’s theorem[24, 25] and design-ing the algorithms for LHV models[26, 27]In this work, we investigate the hierarchy of entan-glement and nonlocality, considering arbitrary projectivemeasurements, in the GHZ-symmetric states (GHZ is forGreenberger-Horne-Zeilinger) [28, 29], by using the opti-mal LHS model for Bell diagonal states [30, 31]. TheGHZ-symmetric states are a two-parameter family ofthree-qubit states sharing the symmetries of the GHZstate. Eltschka and Siewert [28] gave a complete descrip-tion of their entanglement class with respect to stochasticlocal operations and classical communication (SLOCC)[32, 33]. The SLOCC classification is more detailed thanthe scheme according to the number of parties, for in-stance the GME in three-qubit states is divided into Wand GHZ classes. One should expect interesting andnovel phenomena due to the subtle structure of entangle-ment. In particular, a natural question is here whethera specific type of GME, W or GHZ, can serves as a suf-ficient condition for GMNL.The optimal LHS models for Bell diagonal states pro-vide a concise criteria for EPR steering for such familyof two-qubit states [30]. Our technique in this work canbe divided into two steps. First, by replacing the hid-den state in the mentioned optimal model with a two-qubit state, we construct a local model for a tripartitestate. Generally speaking, it is difficult to determine theentanglement class of the tripartite state. Second, weperform the GHZ symmetrization [28, 29] on the tripar-tite state and its local model simultaneously, and therebyobtain a GHZ-symmetric state admitting a LHV model.We present a bilocal model and a fully LHV model forthe GHZ-symmetric states, which show entantlement andnonlocality at the same level are inequivalent in general.And the bilocal model offers a nagetive answer to theabove question that, neither the W nor GHZ entangle-ment can serve as a sufficient condition for the genuinetripartite nonlocality.
II. TWO-QUBIT GHZ-SYMMETRIC STATES
Let us begin with the two-qubit GHZ-symmetric statesintroduced by Siewert and Eltschka [29]. Although theLHV models we present in this part is trivially reformu-lated from the the existed LHS models, the following re-sults will serve as important tools for the tripartite case.The GHZ-symmetric states of a two-qubit system sharethe symmetries of the two Bell states | φ ± i = √ ( | i ±| i ) and can be written as ρ S = ( √ q + p ) (cid:12)(cid:12) φ + (cid:11) (cid:10) φ + (cid:12)(cid:12) + ( √ q − p ) (cid:12)(cid:12) φ − (cid:11) (cid:10) φ − (cid:12)(cid:12) + (1 − √ q ) , (1)where the two parameters − √ ≤ q ≤ √ and q √ ± p + ≥ p, q ) as shown in Fig. 1. The separableboundary are two lines of q = √ ± √ p . The triangle isactually equivalent to a cross-section of the tetrahedronfor Bell diagonal states in the space of ( T x , T y , T z ) [34,35]. Here, the Bell diagonal states are of the form ρ B = 14 [ T ~σ ) · ~σ ] , (2)whose spin correlation matrix is diagonal as T =Diag[ T x , T y , T z ], with ~σ = ( σ x , σ y , σ z ) T being the vec- - - - - - p q FIG. 1: Triangle of two-qubit GHZ-symmetric states, con-strained by q √ ± p + ≥ − √ ≤ q ≤ √ . The uppercorners of the triangle are the Bell state | φ + i and | φ − i . Themaximally mixed state 1 locates at the origin. The entan-gled (red) and separable (yellow) regions are divided by thelines of q = √ ± √ p . The black solid curve corresponds tothe LHV model in (8) and satisfies the function (9). tor of Pauli operators. The matrices of GHZ-symmetricstates (1) satisfy T x = − T y = 2 p and T z = 2 √ q .The optimal LHS models for Bell diagonal states areoriginally constructed by Jevtic et al. [30] based onthe steering ellipsoid [36]. These models draw an EPR-steerable boundary in the space of ( T x , T y , T z ). Zhangand Zhang [31] show a simple approach to generate themfrom Werner’s results. We here follow the formulae inthe latter work.Let ρ denote a state of the whole system of A and B and Π ~xa be a projector of an measurement labeled by ~x corresponding to outcome a . After a local measurementon subsystem A , the unnormalized postmeasured state of B is ρ ~xa = Tr A (Π ~xa ⊗ ρ ) , where Tr A is the partial traceover A . A LHS model is defined as ρ LHS = Z ω ( ~λ ) P A ( a | ~x, ~λ ) ρ λ d~λ. (3)where ~λ represents a hidden variable with a distribution ω ( ~λ ), ρ λ is a hidden state depending on ~λ , and P A ( a | ~x, ~λ )is the probability of outcome a under the condition of ~x and ~λ . If there exists a LHS model satisfying ρ ~xa = ρ LHS , for all the measurements, the state ρ is unsteerable from A to B .Suppose that T is the spin correlation matrix of aBell diagonal state on the EPR-steerable boundary. Theprojector for a qubit can be expressed as Π ~xa = ( a ~x · ~σ ) with ~x being a unit vector and a = ±
1. Thecorresponding postmeasured states can be derived as ρ ~xa = 14 [ a ( T ~x ) · ~σ ] . (4)In the optimal LHS models, the hidden variable is a vec-tor ~λ on a unit sphere. Its distribution, the conditionalprobability and the corresponding hidden state are givenby ω ( ~λ ) = | T ~λ | π ,P A ( a | ~x, ~λ ) = 12 [1 + a sgn( ~x · ~λ )] , (5) ρ λ = 12 ( ~λ ′ · ~σ ) , with ~λ ′ = T ~λ/ | T ~λ | . Substituting them into the integral(3), one can find that they satisfy ρ ~xa = ρ LHS , when thenormalization condition Z | T ~λ | d~λ = 2 π, (6)is fullfilled. And the EPR-steerable boundary is deter-mined by the normalization condition (6). One can triv-ially construct a LHV model here by introducing the re-sponse function for part BP B ( b | ~y, ~λ ) = Tr(Π ~yb ρ λ ) , (7)where the projection operator Π ~yb has the similar defini-nation as Π ~xa . Then, the outcomes of local measurementscan be simulated by the LHV model as P ( a, b | ~x, ~y ) = Tr(Π ~xa ⊗ Π ~yb ρ B )= Z ω ( ~λ ) P A ( a | ~x, ~λ ) P B ( b | ~y, ~λ ) d~λ. (8)We can explicitly solve the integral for the GHZ-symmetric states (1) and express it as1 | p | = | w | + arccosh | w |√ w − . (9)with w = √ q/p . It is the EPR-steerable boundary onthe plane of ( p, q ) displayed as the black curve in Fig. 1.The states between the curve and the separable bound-ary can be sufficiently determined to be entangled butwithout nonlocality. III. THREE-QUBIT GHZ-SYMMETRIC STATES
We now turn to the three-qubit GHZ-symmetric states.These states share the following symmetries of the twoGHZ states | G ± i = √ ( | i ± | i ): (i) qubit permuta-tions, (ii) simultaneous three-qubit flips (i.e., applicationof σ x ⊗ σ x ⊗ σ x ), (iii) qubit rotations about the z axisof the form U ( φ , φ ) = e iφ σ z ⊗ e iφ σ z ⊗ e − i ( φ + φ ) σ z . - - - p q FIG. 2: Triangle of tripartite GHZ-symmetric states. Thestandard GHZ states | G ± i are located at the two uppercorners, and the maximally mixed state 1 at the origin.Different classes of entanglement (separable, biseparable, Wand GHZ) are indicated from yellow to red in order. Theirboundarie in order are defined by: | p | = − √ q + (separa-ble/biseparable), | p | = − √ q + (biseparable/W), and theparametric curve with p = ( v + 8 v ) / [8(4 − v )] and q =[ √ − v − v )] / [4(4 − v )] (W/GHZ). The black solid curvecorresponds to ρ S [ ρ ( T )], which admits a bilocal model, andsatisfies the equation (9) but with w = (2 √ q − / / | p | .The dashed curve corresponds to ρ S [ ρ ( T )], which admits afully LHV model, and satisfies the parametric equation (23). They are of the form ρ S = (cid:18) q √ p (cid:19) | G + i h G + | + (cid:18) q √ − p (cid:19) | G − i h G − | + (cid:18) − q √ (cid:19) , (10)where the two parameters satisfy − √ ≤ q ≤ √ and | p | ≤ + √ q . As shown in Fig. 2, this family of statesforms an isosceles triangle in the state space. Two purestates | G ± i locate at the two upper corners, and origincorresponds to the maximally mixed state . Eltschkaand Siewert [28] divide the triangle into four regionsaccording the entanglement classes as fully separable,biseparable, W, or GHZ, which are displayed by differentcolors in Fig. 2.For an arbitrary three-qubit state ρ , one can derivea corresponding GHZ-symmetric state by performing aGHZ symmetrization ρ S ( ρ ) = Z du ( uρu † ) , (11)where the integral is to cover the entire GHZ symmetrygroup, i.e., the operations in (i), (ii), (iii) and their prod-ucts. The coordinates of ρ S ( ρ ) can be inferred from twomatrix elements of ρ as p = 12 (cid:18) h G + | ρ | G + i−h G − | ρ | G − i (cid:19) ,q = 1 √ (cid:18) h G + | ρ | G + i + h G − | ρ | G − i− (cid:19) . (12)We make a remark here that, to obtain a state ρ S ( ρ ), the state ρ does not need to be a state. Namely, the oper-ator ρ is required to be Hermitian and normalized, butthe positive semi-definitiveness can be replaced with thecondition that its corresponding ( p, q ) are in the physicalregion. In the following parts, we actually utilize such states to construct local GHZ-symmetric states.The nonlocality (GMNL, bilocality or full locality) of ρ is preserved by the operations in (i), (ii) and (iii), whichare local unitary or permutations. Consequently, theGHZ symmetrization only could reduce or preserve thenonlocality. Here to reduce means to change the non-locality into a weaker one, e.g., to change GMNL intobilocality or full locality. Said a different way, if the state ρ admits a LHV model, by symmetrizing the model, wecan construct a one for ρ S ( ρ ). Based on this idea and theresults in two-qubit case, we given the following modelsfor ρ S and sufficient criterions for its locality. A. Bilocal model
A direct result can be obtained by replacing the basis {| i , | i} of B in two-qubit GHZ symmetric states with {| i , | i} of system B and C . This leads to a familyof three-qubit states for ABCρ ( T ) = 14 h ⊗ Σ + ( T ~σ ) · ~ Σ i , (13)where Σ = | ih | + | ih | , Σ x = | ih | + | ih | , Σ y = − i | ih | + i | ih | and Σ z = | ih | −| ih | are the generalized Pauli matrices in the sub-space of {| i , | i} . When T = T satisfy the normal-ization condition (6) and thereby on the EPR-steerableboundary, The tipartite states naturally admit a familyof LHV models as P b ( a,b,c | ~x, ~y, ~z ) = Tr h Π ~xa ⊗ Π ~yb ⊗ Π ~zc ρ ( T ) i = Z ω ( ~λ ) P A ( a | ~x,~λ ) P BC ( b,c | ~y,~z,~λ ) d~λ, (14)where P BC ( b, c | ~y, ~z, ~λ ) = Tr(Π ~yb ⊗ Π ~zc ρ BCλ ), ρ BCλ = (Σ + ~λ ′ · ~ Σ), and other parameters and functions are defined in(20). Since the bipartite state ρ BCλ is an entangled purestate in general, the response function P BC ( b, c | ~y, ~z, ~λ )can not be further decomposed. The models are bilocal,which exclude GMNL in the states but allow bipartitenonlocality between B and C .One can now symmetrize the states ρ ( T ) into the p E n t ang l e m en t FIG. 3: Entanglement degrees vs. the p coordinate of ρ S [ ρ ( T )] . The curves show the amounts of total entan-glement, GME and GHZ entanglement in order from top tobottom. GHZ symmetric family. Actually, only the permuta-tions are necessary, as the states are invariant underthree-qubit flips and the rotations U ( φ , φ ). The corre-sponding GHZ-symmetric states ρ S [ ρ ( T )] draw a curve,which can also be parameterized as the equation (9) butwith w = ( √ q − / /p , on the plane of ( p, q ) as shownin Fig. 2. They admit a bilocal model which can beexpressed as an average of the one in (14) and its twopermutations of A ↔ B and A ↔ C . We omit the for-mula here.The convex combinations of ρ S [ ρ ( T )] and fullyseparable states form the region under the curve for ρ S [ ρ ( T )] in the triangle, in which the GMNL is cer-tainly excluded. It is worth noting that curve passesthrough both W and GHZ entanglement, although it isjust marginally higher than the W/GHZ boundary nearthe state ρ S = ( | G + ih G + | + | G − ih G − | ). Consequently,both the two classes of genuine three-qubit entanglement,W and GHZ, are inequivalent to genuine three-qubit non-locality.One can observe the deviations of ρ S [ ρ ( T )] from theentanglement boundaries by deriving its entanglementdegrees, shown in Fig. 3. We adopt three measures forthe entanglement in GHZ-symmetric states. The three-tangle is a natural choice to quantify the GHZ entan-glement, as Siewert and Eltschka [29] have derived itsanalytical results for the GHZ-symmetric states. Givena state with coordinates ( p, q ) ( p ≥
0, without loss ofgenerality), its three-tangle is given by τ ( p, q ) = max (cid:26) , p − p W − p W (cid:27) (15)where p W is the p coordinate of the intersection pointof the straight line, connecting ( p, q ) and the corner for | G + i , and the W/GHZ boundary. The GHZ-symmetricstates belong to the family of three-qubit X matrices.Their total entanglement (opposite to full separability)and GME can be measured by two generalizations ofconcurrence [37, 38], which can be expressed by usingthe coordinates asC T = max ( , | p | + √ q − ) , C G = max (cid:26) , | p | + √ q − (cid:27) . (16)Similiar as the three-tangle (15), both of them can be ge-ometrically interpreted as a relative distance to its cor-responding boundary. As shown in Fig. 3, the GHZentanglement has a nonzero section near its left end-point. The other two degrees of entanglement are alwaysgreater than zero, and total entanglement monotonouslyincreases with ρ S [ ρ ( T )] moving from the top line of thistriangle to the right side. B. Fully local model
Our approach can be extended to construct a fullyLHV model, by modifying the hidden state ρ BCλ to alocal one. That is, when ρ BCλ admits a local model as P BC ( b, c | ~y, ~z, ~λ ) = Tr(Π ~yb ⊗ Π ~zc ρ BCλ )= Z τ ( ~µ,~λ ) P B ( b | ~y, ~µ, ~λ ) P C ( c | ~z, ~µ, ~λ ) d~µ, with a hidden variable ~µ and a distribution τ ( ~µ, ~λ ), theintegral in (14) presents a fully LHV model as P f ( a,b,c | ~x, ~y, ~z ) = Z ω ( ~λ ) P A ( a | ~x,~λ ) P BC ( b,c | ~y,~z,~λ ) d~λ. (17)Here, the hidden variable is ( ~λ, ~µ ) in a joint distributionΩ( ~λ, ~µ ) = ω ( ~λ ) τ ( ~µ, ~λ ).To analytically construct a fully local model for GHZ-symmetric states, we extend the three-qubit states (13)to the form ρ ( T ) = 14 h t ⊗ Σ +( T ~σ ) · ~ Σ i + 12 (1 − t ) ⊗ D , (18)where t ∈ [0 ,
1] and D is a normalized diagonal state inthe subspace of {| i , | i} . The coordinates of ρ S [ ρ ( T )]are determined by the first part of ρ ( T ) as p = 12 T x , q = 12 √ (cid:18) t + T z − (cid:19) . (19)We now present a one-parameter family of fully LHVmodels corresponding to a family of states in above form.Suppose that the paremeter and matrix in the states are t = t and T = T . The hidden variable is still choose asa vector ~λ on a unit sphere. We define its distribution,the conditional probability and the corresponding hidden state as ω ( ~λ ) = | T ~λ | π h q ( ~λ ) i ,P A ( a | ~x, ~λ ) = 12 h a sgn( ~x · ~λ ) i , (20) ρ BCλ = 11 + q ( ~λ ) (cid:20) (cid:16) Σ + ~λ ′′ · ~ Σ (cid:17) + q ( ~λ ) D (cid:21) , where ~λ ′′ = T ~λ/ | T ~λ | , q ( ~λ ) ∈ [0 ,
1] and q ( − ~λ ) = q ( ~λ ).Substituting them into (17) and requiring P f ( a,b,c | ~x, ~y, ~z ) = Tr h Π ~xa ⊗ Π ~yb ⊗ Π ~zc ρ ( T ) i , (21)one can obtain the relations Z | T ~λ | π d~λ = t , Z | T ~λ | π q ( ~λ ) d~λ = 1 − t . (22)Therefore, a given form of q ( ~λ ) can determine a curve onthe triangle. We choose q ( ~λ ) = min { sin θ ′′ , c = 1 − w c } ,where θ ′′ is the angle between the positive z-axis and ~λ ′′ ,and w c ≈ .
354 is the value of w on the intersection pointof the curve (9) and the side of the triangle for two-qubitcase. The definition of D corresponding to the two op-tions is D = ( | ih | + | ih | ) for q ( ~λ ) = sin θ ′′ and D = cos θ ′′ | ih | + sin θ ′′ | ih | for q ( ~λ ) = 1 − w c .In the first case, ρ BCλ is separable. In the second case, ρ BCλ is unsteerable, as it is equivalent to the state on theabove-mentioned intersection point, under the local posi-tive linear map M = ( e iφ ′′ sin θ ′′ | ih | + cos θ ′′ | ih | ) ⊗ q ( ~λ ) is a piecewise function. Direct calculationgives the coordinates of ρ S [ ρ ( T )] as a parametric equa-tion p = 12 G ( v ) , q = 12 √ (cid:20) G ( v ) + vG ( v ) − (cid:21) , (23)where G ( v ) = G ( v ) + G ( v ), and G ( v ) and G ( v ) aretwo functions of v , given by G ( v ) = v arctan √ − v √ − v + 1 ,G ( v ) = c v p c ( v −
1) + 12 [ c ( v −
1) + 1] + c arccosh q c ( v − c ( v − √ v −
1+ arctan √ − c cv + cv √ − c c ( v −
1) + 1 . The one-parameter family of states ρ S [ ρ ( T )] draws acurve on the triangle, the region under which are of fullylocal states. This shows that the bipartite entanglementis not equivalent to bipartite nonlocality in the GHZ-symmetric states. IV. SUMMARY
We investigate the relations between entanglementand nonlocality in multipartite systems considering theSLOCC classification of entanglement. However, bothdetermining entanglement properties and constructinglocal models for a given set of multipartite states aretwo challenges in theoretic studies. We adopt a two-parameter family of three-qubit states, the so calledGHZ-symmetric states, as a research object, as the en-tanglement of these states have an exact description. Byperforming the GHZ symmetrization on the tripartite ex-tensions of two-qubit local model, we present two one-parameter families of LHV models. These models showthat both the bipartite entanglement and genuine tripar-tite entantlement are inequivalent to the nonlocality atthe same level. In particular, neither the W GME nor theGHZ GME can serve as a sufficient condition for genuine tripartite nonlocality.A natural open question left here is that whether boththe W and GHZ type entangled states can admit a fullylocal model. One can try to symmetrize the fully localstate ρ F constructed by Bowles et al. [16] with N = 3into the GHZ-symmetric form, however the result is aseperable state. Development of new methods to con-struct fully LHV models for GHZ-symmetric states withGME is still necessary. An alternative approach may beto determine the entanglement class for a fully local state,such as the mentioned ρ F and ρ ( T ) in our results. Theapproach of the convex characteristic curve [29, 40] of-fers a possible solution to classify the genuine tripartiteentantlement by calculating the amount of three-tangle. Acknowledgments
This work was supported by the NSF of China (GrantsNo. 11675119, No. 11575125, and No. 11105097). [1] M. A. Nielsen and I. L. Chuang,
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