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QuantumInformationReview

April2011 Vol.56 No.10: 945-954

doi: 10.1007 s11434-011-4395-1 c

The Author(s) 2011. This article is published with open access at Springerlink.comcsb.scichina.com www.springer.com/scpBell inequality, separability and entanglement distillation

LI Ming

1, FEI ShaoMing2,3*& LI-JOST XianQing3,4

1College of Mathematics and Computational Science, China University of Petroleum, Dongying 257061, China;

2School of Mathematical Sciences, Capital Normal University, Beijing 100048, China;

3Max-Planck-Institute for Mathematics in the Sciences, Leipzig 04103, Germany;

4Department of Mathematics, Hainan Normal University, Haikou 571158, China

Received August 3, 2010; accepted October 10, 2010In

this review, we introduce well-known Bell inequalities, the relations between the Bell inequality and quantum separability, and the

entanglement distillation of quantum states. It is shown that any pure entangled quantum state violates one of Bell-like inequalities.

Moreover, quantum states that violate any one of these Bell-like inequalities are shown to be distillable. New Bell inequalities that

detect more entangled mixed states are also introduced. Bell inequality, entanglement, distillationCitation:Li

M, Fei S M, Li-Jost X Q. Bell inequality, separability and entanglement distillation. Chinese Sci Bull, 2011, 56: 945-954, doi: 10.1007

s11434-011-4395-1The contradiction between local realism and quantum me- chanics was first highlighted by the paradox of Einstein, Podolsky and Rosen (EPR) [1]. Nonlocality can be deter- mined from violation of conditions, called Bell inequalities [2], that are satisfied by any local variable theory. In 1964, Bell formulated an inequality that is obeyed by any local hidden-variable theory. However, he showed that the EPR singlet statej +i=1p2 j 00 i+j11i) violates the inequality. In fact, the Bell inequality provided the first possibility to distin- guish experimentally between quantum-mechanical predic- tions and predictions of local realistic models. Bell inequal- ities are of great importance in understanding the concep- tual foundations of quantum theory and investigating quan- tum entanglement, as they can be violated by quantum entan- gled states.On the other hand, violation of the inequalities is closely related to the extraordinary power of realizing certain tasks in quantum information processing, which outperforms its classical counterpart, such as building quantum protocols to decrease communication complexity [3] and providing se- cure quantum communication [4, 5]. One of the most important Bell inequalities is the Clauser- Horne-Shimony-Holt (CHSH) inequality [6] for two-qubit*Corresponding author (email: feishm@mail.cnu.edu.cn) systems. It can be generalized to theN-qubit case, known as the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality [7-9]. A set of multipartite Bell inequalities has been ele- gantly derived in terms of two dichotomic observables per site [10, 11]. The set includes the MABK inequality as a spe- cial case [12] and can detect entangled states that the MABK inequality fails to detect. Ref. [13] introduced another fam- ily of Bell inequalities forN-qubit systems that are maxi- mally violated by all Greenberger-Horne-Zeilinger states. A method of extending Bell inequalities fromnto (n+1)-partite states is described in [14]. In the higher dimensional bipartite case, Collins et al. [15] constructed a CHSH-type inequal- ity for arbitraryd-dimensional (qudit) systems known as the Collins-Gisin-Linden-Masser-Popescu (CGLMP) inequality. Gisin [16] presented a theorem in 1991 that states that any pure entangled two-qubit state violates the CHSH inequal- ity. Specifically, the CHSH inequality is both su cient and necessary for the separability of two-qubit states. Soon af- ter, Gisin and Peres [17] provided an elegant proof of this theorem for the case of pure two-qudit systems. Chen et al. [18] showed that all pure entangled three-qubit states violate a Bell inequality. Nevertheless, it has remained an open prob- lem for a long time whether Gisin"s theorem can be general- ized to the multi-qudit case. In addition, Bell inequalities that

946LiM,et al. Chinese Sci BullApril (2011) Vol.56 No.10

can detect more (mixed) entangled quantum states are being searched for. Bell inequalities are also useful in verifying the security of quantum key distribution protocols [19, 20]. There is a simple relation between nonlocality and distillability: if any two-qubit [21] or three-qubit [22] pure or mixed state vio- lates a specific Bell inequality, then the state must be distill- able. D ¨ur showed that for the caseN>8, there existN-qubit bound entangled (non-distillable) states that violate Bell in- equalities [23]. However, Ac

´ın has demonstrated that for all

states violating an inequality, there exists at least one kind of bipartite decomposition of the system such that a pure entan- gled state can be distilled [24, 25]. However generally it is an open problem whether violation of a Bell inequality implies distillability. In this review, we first give a brief introduction of several important Bell inequalities in section 1. We introduce a set of su cient and necessary for the separability of general pure bipartite quantum states in arbitrary dimensions. We then show that pure entangled states can be distilled from quan- tum mixed states that violate one of these Bell inequalities. New Bell operators are constructed in section 3 and used to detect more entangled quantum states. We further derive the maximal violation of such Bell inequalities. We give conclu- sions and remarks in section 4.

1 Some well-known Bell inequalities

In this section we recall several useful Bell inequalities in- cluding the CHSH inequality, WWZB inequality (including the MABK inequality as a special case), CGLMP inequality and some other generalized inequalities.

1.1 Bell inequalities for two and three-qubit systems

The famous CHSH [6] inequality is a kind of improved Bell inequality that is more feasible for experimental verification. Suppose two observers, Alice and Bob, are separated spa- tially and share two qubits. Alice and Bob each measure a dichotomic observable with possible outcomes1 in one of two measurement settings:A1;A2andB1;B2respectively. The CHSH inequality is a constraint on correlations between Alice"s and Bob"s measurement outcomes if a local realistic description is assumed. The Bell function for the CHSH in- equality has been given as [26] B )=A1()(B1()+B2())+A2()(B1()B2());(1) whereis a collection of local hidden variables and the vari- ablesAi() andBj() take values1. According to the lo- cal hidden-variable theory, the statistical average of the Bell function must satisfy the inequality [6, 26],jhB()ij62, where the statistical averagehB()i=R()B()dwith ) the probability density distribution.Quantum mechanically the statistical average of the Bell function is replaced by a quantum average of the correspond- ing operator given by B=A1 B1+A1 B2+A2 B1A2

B2;(2)

whereAi=~ai~A=axixA+ay iy A+az iz

A,Bj=~bj~B=

b xjxB+by jy B+bz jz

B,~ai=(axi;ay

i;az i) and~bj=(bxj;by j;bz j) are real unit vectors satisfyingj~aij=j~bjj=1 withi;j=1;2, and x;y;z A =Bare Pauli matrices. The CHSH inequality says that if there exist local hidden-variable models to describe the sys- tem, the inequality jhBij62(3) must hold. For entangled states, it is always possible to find suitable observablesA1,A2,B1andB2such that inequality (3) is vio- lated. For instance, takingj +i=(j01i j10i)=p2,A1=x, A

2=z,B1=(x+z)=p2,

andB2=(xz)=p2, we obtainjhBij=2p2, which gives the maximal violation [27]. For three-qubit states, the Mermin inequality states that [7-9] jhA2B1C1i+hA1B2C1i+hA1B1C2i hA2B2C2ij62;(4) where observablesAi;Bi;andCi,i=1;2, are associated with three qubits respectively. The maximal violation of the inequality (4) is 4. The quantum mechanical violation of the Bell inequalities has been demonstrated experimentally, e.g. [28].

1.2 Bell inequalities for multipartite qubit systems

The MABK inequality is a kind of Bell inequality for multi- partite qubits [7-9] whereas the WWZB inequality [10, 11] is a kind of generalization of the MABK inequality. Here we introduce the WWZB inequality and consider the MABK inequality as a special case of the WWZB inequality. Consider anN-qubit quantum system and allow each part to choose independently between two dichotomic observ- ablesAj;A0 jfor thejth observer, specified by local parame- ters. Each measurement has two possible outcomes 1 and1. The WWZB quantum mechanical Bell operator is defined by B N=12 NX s

1;s2;;sN=1S(s1;s2;

;sN) X k

1;k2;;kN=1sk11sk22skNN

N j

1Oj(kj);(5)

whereS(s1;s2;;sN) is an arbitrary function taking only values1 andOj(1)=AjandOj(2)=A0 jwithkj=1;2:It is shown in [10, 11] that local realism requiresjhBNij61: The MABK inequality is recovered by takingS(s1;s2;; s N)=p2 cos[(s1+s2++sNN+1)=(¼=4)] in (5). Employing an inductive method from the (N1)-partite WWZB Bell inequality to theN-partite inequality, a family LiM,et al. Chinese Sci BullApril (2011) Vol.56 No.10947 of Bell inequalities was presented in [13]. The new Bell op- erator is defined by B N=BN1 12 (AN+A0

N)+IN1

12 (ANA0

N);(6)

whereBN1represents the normal WWZB Bell operators defined in (5). Such new Bell operators yield violation of the Bell inequality for the generalized GHZ states,j i= cosj000i+sinj111i, in the whole parameter region ofand for any number of qubits, thus overcoming the draw- back of the WWZB inequality. In the three-qubit case, one can construct three di erent Bell operators fromB2by taking theapproachof(6). ThecorrespondingthreeBellinequalities can distinguish full separability, detailed partial separability and true entanglement [29].

1.3 Bell inequalities for high-dimensional systems

For bipartite high-dimensional quantum systems, we intro- duce the CGLMP inequality given in [15]. We consider the standard Bell-type experiment: two spatially separated ob- servers, Alice and Bob, share a copy of a pure two-qudit state j i 2Cd

Cdin the composite system. Suppose that Alice

and Bob both have the choice of performing one of two dif- ferent projective measurements, each of which hasdpossible outcomes. LetA1andA2denote observables measured by Alice andB1andB2the observables measured by Bob. Each measurement hasdpossible outcomes: 0;1;;d1. Any local variable theory must then obey the well-known CGLMP inequality [15]: I d[ d2 1X k=0(12kd1)f[P(A1=B1+k) +P(B1=A2+k+1)+P(A2=B2+k) +P(B2=A1+k)][P(A1=B1k1) +P(B1=A2k)+P(A2=B2k1) +P(B2=A1k1)]g62:(7) Here [x] denotes the integer part ofx. The joint probability P (Aa=Bb+m)=Pd1 j

0P(Aa=j;Bb=jm),a;b=1;2, in

which the measurementsAaandBbhave outcomes that dier bym(modd). Chen et al. show that all bipartite entangled states violate the CGLMP inequality [30], which gives a detailed proof of

Gisin"s Theorem for two-qudit quantum systems.

LetX[1]

jandX[2] j, wherej=1;2, denotes the two ob- servables for thejth party. Each hasdpossible outcomes: x [1] j;x[2] j=0;1;;d1. Fu introduced the correlation func- tionQij[31], Q ij=1S d1X mquotesdbs_dbs15.pdfusesText_21