Entanglement detection via the condition of quantum correlation

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Entanglement detection via the condition of quantum correlation

Che-Ming Li,1,2Li-Yi Hsu,3Yueh-Nan Chen,4Der-San Chuu,1,*and Tobias Brandes5

1_{Department of Electrophysics, National Chiao Tung University, Hsinchu 30050, Taiwan} 2

Physikalisches Institut, Universität Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany 3

Department of Physics, Chung Yuan Christian University, Chung-li 32023, Taiwan 4

Department of Physics and National Center for Theoretical Sciences, National Cheng Kung University, Tainan 701, Taiwan 5

Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36 D-10623 Berlin, Germany 共Received 18 May 2007; published 12 September 2007兲

We develop a necessary condition of quantum correlation. It is utilized to construct a d-level bipartite Bell-type inequality which is strongly resistant to noise and requires only analyses of O共d兲 measurement outcomes compared to the previous result O共d2_{兲. Remarkably, a connection between the arbitrary} high-dimensional bipartite Bell-type inequality and entanglement witnesses is found. Through the necessary condi-tion of quantum correlacondi-tion, we propose that the witness operators to detect truly multipartite entanglement for a generalized Greenberger-Horne-Zeilinger共GHZ兲 state with two local measurement settings and a four-qubit singlet state with three settings. Moreover, we also propose a robust entanglement witness to detect a four-level tripartite GHZ state with only two local measurement settings.

DOI:10.1103/PhysRevA.76.032313 PACS number共s兲: 03.67.Mn, 03.65.Ud Entanglement is at the heart of quantum physics and a

resource for quantum information processing 关1兴.

Multipar-tite entanglement for two-level quantum systems共qubits兲 has attracted attention for its unusual features 关2兴 and necessity

in a large-scale realization of quantum computation and com-munication关3兴. In particular, with the rapid development of

technology for manipulating quantum states, multipartite en-tanglement has been created experimentally and then utilized for quantum information processing 关4兴. In addition,

en-tangled qubits, entanglement for multilevel quantum systems 共qudits兲 has been realized in a few physical systems 关5兴.

Moreover, it has been proven that qudits have an advantage over qubits关6兴. Thus, identifying whether an experiment’s

output is an entangled state for multipartite or multilevel systems is very important for further studies on quantum correlation and to perform reliable quantum protocols.

Bell-type inequalities共BIs兲关7–9兴 and entanglement

wit-nesses 共EWs兲关10–13兴 are widely used to verify quantum

correlation. BIs are based on the local hidden variable theo-ries whereas EWs rely on an utilization of the whole or par-tial knowledge of the entangled state to be created. However, a single systematic approach to construct EWs for entangled qudits and to connect BIs for arbitrary high-dimensional sys-tems with EWs is still lacking. Investigations on how en-tangled qudits can be shown efficiently and what the funda-mental feature is in entanglement verifications are both significant for a deeper understanding of quantum correlation of qudits 关14兴 and for efficient manipulations to achieve

quantum information processing关15兴.

In this work, we develop a necessary condition of quan-tum correlation. This enables d-level bipartite BIs to be tested with only analyses of O共d兲 measurement outcomes for detection events which is much smaller than the previous result O共d2_{兲关}₉_,₁₆_{兴. In particular, a connection between}

arbi-trary high-dimensional bipartite BIs and EWs is found. We

then use the correlator operators involved in the necessary condition of quantum correlation to construct EWs for de-tecting genuine multipartite entanglement, which can only be generated with participation of all parties of a system, in the generalized Greenberger-Horne-Zeilinger 共GHZ兲 state with two local measurement settings共LMSs兲共which will be de-scribed in detail兲 and four-qubit singlet states 关17兴 with only

three LMSs. More recently, it has been shown that the four qubit singlet state is very useful for quantum secret sharing 关18兴. Through our method, the 15 LMSs required for the EW

by Ref. 关12兴 can be reduced greatly. In order to show the

high generality of the condition of quantum correlation, we also describe an EW that can detect a four-level tripartite GHZ state关14兴 with only two LMSs. Moreover, the proposed

EWs are resistant to noise. In what follows, an introduction to the necessary condition of quantum correlation will be given as a preliminary to further applications.

I. CORRELATION CONDITIONS FOR QUANTUM CORRELATION

In an experiment whose aim is to generate a multipartite entangled state兩␰典, if the experimental conditions are imper-fect, it is important to know whether an experimental output state still possesses multipartite quantum correlation which is close to the state兩␰典. One EW for detecting genuine multi-partite entanglement is given by Ref.关11兴 and formulated as

W_␰p₌_␣ ␰

p_{1 −}_兩_␰_典具_␰_兩, _共1兲

where␣_␰p= max_{兩␹典僆B}兩具␹兩␰典兩2and B denotes the set of bisepa-rable states. Although it is difficult to determine the overlap

␣␰p, through the general method proposed by Bourennane et

al. 关12兴, one can perform this task. Thus, for some

experi-mental output state, say␳, if measured outcomes show that Tr共W_␰p_␳_{兲⬍0, the state}_␳

is identified as a genuine multipar-tite entanglement which is close to the state兩␰典.

It is worth noting that complete knowledge of the state 兩␰典, i.e., all information about correlation characters, is uti-*Electronic address: dschuu@mail.nctu.edu.tw

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lized for the witness operator and in order to measure the operatorW_␰pexperimentally, the number of LMSs appears to increase with the number of qubits of the state 兩␰典关12兴. A

LMS, denoted by M :共Vˆ1, . . . , Vˆn兲 in this paper, means that

single-qubit measurements of operator Vˆi for i = 1 , . . . , n are taken on the n remote parties in parallel. In addition, EWs with forms such as W_␰p, the number of LMSs utilized to realize BIs typically increases exponentially with the number of parties of the state. Moreover, the analyses of measured outcomes for detection events also depend on the structures of BIs. A detection event means a set of measurement out-comes, denoted by 共v1, . . . ,vn兲, under some LMS. For

ex-ample, the LMS M2z=共␴z,␴z兲 corresponds to four possible detection events:共0,0兲, 共0,1兲, 共1,0兲, and 共1,1兲, where vi= 0 or 1 stands for the eigenvalue共−1兲vi _{of Pauli operator}␴

z. The meaning of LMS and that of the detection event are strictly different.

The witness operators proposed in this paper to detect genuine multipartite entanglement have the following form:

W_␰=␣_␰1 − Cˆ_␰, 共2兲

where␣_␰ is some constant and Cˆ_␰ is the operator which is composed of several different kinds of correlator operators with necessary conditions of quantum correlations imbedded in the state 兩␰典. If outcomes of measurements show that Tr共W_␰␳兲⬍0, the state ␳ is identified as a truly multipartite entanglement. In what follows we will show that the operator Cˆ_␰ can be constructed systematically and measured with fewer LMSs for different kinds of pure multipartite en-tangled qubits or qudits.

Furthermore, through the same idea behind the method to construct correlator operators, a d-level bipartite BI is con-structed and able to be tested experimentally with fewer analyses of detection events. We then consider the correla-tion condicorrela-tions for quantum correlacorrela-tion involved in the ap-proach to construct correlator operators utilized in EWs and BIs as a connection between them. We will see that the building blocks of the proposed EWs and BIs are all derived from the correlation conditions for quantum correlation.

In order to present the idea behind the correlation condi-tion for quantum correlacondi-tion clearly, let us first illustrate a derivation of correlation condition for the generalized four-qubit GHZ state

兩⌽共␪,␾兲典 = cos共␪兲兩0000典z+ ei␾sin共␪兲兩1111典z 共3兲 for 0⬍␪⬍␲/ 4 and 0ⱕ␾⬍␲/ 2, where 兩v1v2v3v4典z=

丢k=14 兩v典kzforv僆兵0,1其 and 兩v典kzcorresponds to an eigenstate of␴zwith eigenvalue共−1兲vfor the party k. For the four-qubit system, the kernel of our strategy for identifying correlation between a specific subsystem, say A, and another one, say B, under some LMS, Ml, relies on the sets of correlators with the following forms:

C₀共l兲= P共vA0,vB0兲 − P共vA1,vB0兲, 共4兲

C₁共l兲= P共vA1,vB1兲 − P共vA0,vB1兲, 共5兲 where P共vAi,vBj兲 is the joint probability for obtaining the measured outcomesvAifor the A subsystem andvBjfor the B one. By the values of the correlators for an experimental output state, we could identify correlations between out-comes of measurements for the subsystems.

Proposition 1. If the results of measurements reveal that C₀共l兲 and C₁共l兲 are all positive or all negative, i.e., C₀共l兲C₁共l兲⬎0, we are convinced that the outcomes of measurements formed on the A subsystem are correlated with the ones per-formed on the B subsystem.

Proof. If the A subsystem is independent of the B one, we recast P共vAi,vBj兲 as P共vAi兲P共vBj兲, where P共vAi兲 and P共vBj兲 denote the marginal probabilities for obtaining resultsvAiand vBj, respectively. Then, we have

C_0,n共l兲 =关P共vA0兲 − P共vA1兲兴P共vB0兲, 共6兲 C_1,n共l兲 =关P共vA1兲 − P共vA0兲兴P共vA1兲. 共7兲 Since P共vA1兲, P共vB0兲ⱖ0, we conclude that C₀共l兲C₁共l兲ⱕ0. Therefore, C₀共l兲C₁共l兲⬎0 implies that the measured outcomes performed on the A subsystem are dependent on the one performed on the B subsystem. Q.E.D.

We start showing the strategy with the help of proposition 1. First, to describe the correlation between a specific party and others of the four-qubit system, we give four sets of correlator operators

Cˆ_0,nz共z兲 =共0ˆnz− 1ˆnz兲丢0ˆmz丢0ˆpz丢0ˆqz, 共8兲 Cˆ1,nz共z兲 =共1ˆnz− 0ˆnz兲丢1ˆmz丢1ˆpz丢1ˆqz, 共9兲

for n = 1 , . . . , 4, wherevˆnz=兩v典nznz具v兩 and n, m, p, and q de-note four different parties under the LMS M_4z =共␴z,␴z,␴z,␴z兲. In order to have compact forms, in what follows, symbols of tensor product will be omitted from cor-relator operators. Then, for some experimental output state, the expectation values of the Hermitian operators Cˆ_0,n共z兲 and Cˆ_1,n共z兲 are expressed in the following correlators in terms of joint probabilities:

C_0,n共z兲 = P共vn= 0,v = 0兲 − P共vn= 1,v = 0兲, 共10兲 C_1,n共z兲 = P共vn= 1,v = 3兲 − P共vn= 0,v = 3兲, 共11兲 wherev =兺_i=1,i4 _⫽nvi. By proposition 1, we know that if results of measurements reveal that C_0,n共z兲C_1,n共z兲⬎0, we are convinced that the outcomes of measurements performed on the nth party are correlated with the ones performed on the rest. If the nth party is independent of the rest, we have

C_0,n共z兲=关P共vn= 0兲 − P共vn= 1兲兴P共v = 0兲, C_1,n共z兲 =关P共vn= 1兲 − P共vn= 0兲兴P共v = 3兲,

and realize that C_0,n共z兲C_1,n共z兲ⱕ0.

For the pure generalized four-qubit GHZ state 兩⌽共␪,␾兲典 we have

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C_{0,n,⌽共␪,␾兲}共z兲 = cos2共␪兲, C_{1,n,⌽共␪,␾兲}共z兲 = sin2共␪兲共12兲 and, hence, C_0,n,共z兲_{⌽共␪,␾兲}C_1,n,共z兲_{⌽共␪,␾兲}⬎0, which describes the out-comes of measurements are correlated. Then the condition C_0,n共z兲C_1,n共z兲⬎0 is a necessary condition of the pure generalized four-qubit GHZ state.

Further, we construct the following correlator operators to identify correlations between a specific group, which is com-posed of the nth and mth parties, and another:

Cˆ_0,nm共z兲 =共0ˆnz0ˆmz− 1ˆnz1ˆmz兲0ˆpz0ˆqz, 共13兲 Cˆ_1,nm共z兲 =共1ˆnz1ˆmz− 0ˆnz0ˆmz兲1ˆpz1ˆqz 共14兲 for n , m = 1 , . . . , 4 and n⫽m. Moreover, we can express the expectation values of the Hermitian operators Cˆ_0,nm共z兲 and Cˆ_1,nm共z兲 in terms of joint probabilities for some output state

vi. Proposition 1 shows that if the subsystem composed of the nth and the mth parties is uncorrelated with another one, the measured out-comes must satisfy C_0,nm共z兲 C_1,nm共z兲 ⱕ0 . On the other hand, C_0,nm共z兲 C_1,nm共z兲 ⬎0 indicates that they are dependent.

It is clear that, for a pure generalized four-qubit GHZ state, we have

C共z兲_0,nm,_{⌽共␪,␾兲}= cos2共␪兲, C共z兲_1,nm,_{⌽共␪,␾兲}= sin2共␪兲, 共17兲 and hence C_{0,nm,⌽共␪,␾兲}共z兲 C_{1,nm,⌽共␪,␾兲}共z兲 ⬎0. Thus we know that the subsystem composed of the nth and the mth parties are cor-related with another. Therefore, the condition, C_0,nm共z兲 C_1,nm共z兲 ⬎0, is also a necessary condition of the state 兩⌽共␪,␾兲典.

After introducing two correlation conditions for the pure generalized GHZ state under M4z, let us progress toward the

third one for correlation. Under the LMS, M4x

=共␴x,␴x,␴x,␴x兲, we formulate four sets of correlators which correspond to the following operators for identifying corre-lations between the nth party and others:

Cˆ_0,n共x兲=共0ˆnx− 1ˆnx兲丢Eˆ , 共18兲 Cˆ_1,n共x兲=共1ˆnx− 0ˆnx兲丢Oˆ , 共19兲 where Eˆ = 共0ˆmx0ˆpx0ˆqx+ 0ˆmx1ˆpx1ˆqx+ 1ˆmx0ˆpx1ˆqx+ 1ˆmx1ˆpx0ˆqx兲, 共20兲 Oˆ = 共1ˆmx1ˆpx1ˆqx+ 1ˆmx0ˆpx0ˆqx+ 0ˆmx1ˆpx0ˆqx+ 0ˆmx0ˆpx1ˆqx兲. 共21兲 From the expectation values of Cˆ_0,n共x兲 and Cˆ_1,n共x兲 for some state and proposition 1, we could know the correlation behavior of the system, i.e., for a system in which the nth party is uncor-related with the rest under M4x, the outcomes of

measure-ments must satisfy the condition C_0,n共x兲C_1,n共x兲ⱕ0.

For the pure state, 兩⌽共␪,␾兲典, the expectation values of Cˆ_k,n共x兲 is given by

C_{0,n,⌽共␪,␾兲}共x兲 = C_{1,n,⌽共␪,␾兲}共x兲 = sin共2␪兲cos共␾兲/2 共22兲 and ensure that there are correlations between measured out-comes under the LMS M_4x. Thus the condition C_0,n共x兲C_1,n共x兲⬎0 is necessary for the pure generalized four-qubit GHZ state.

where␣_⌽共␪,␾兲 is some constant, detects genuine multipar-tite entanglement for the cases 共␪,␾兲: 共␲/ 4 ,␲/ 6兲, 共␲/ 4.9, 0兲, and 共␲/ 3.7,␲/ 9兲. Table I gives a summary of

␣⌽共␪,␾兲 for these cases.

In order to prove that W_⌽共␪,␾兲 is a EW for detecting genuine multipartite entanglement, we have to show the fol-lowing comparison between

W_⌽p_共_␪

,␾兲 =␣_⌽p1 −兩⌽共␪,␾兲典具⌽共␪,␾兲兩共26兲 andW_⌽共␪,␾兲关13兴: if a state␳satisfies Tr关W_⌽共␪,␾兲␳兴⬍0, it also satisfies Tr关W_⌽p共␪,␾兲␳兴⬍0, i.e., W_⌽共␪,␾兲 −␥_⌽W_⌽p共␪,␾兲ⱖ0, where␥_⌽共␪,␾兲 is some positive constant. TABLE I. Summaries of numerical results of ␣_⌽共␪,␾兲 for W⌽共␪,␾兲, the parameters ␥⌽, which are utilized to proveW⌽共␪,␾兲 and␦noise,⌽involved in robustness of the proposed witness operator for detecting truly multipartite entanglement. Three different cases for the state 兩⌽共␪,␾兲典 corresponding to W_⌽共␪,␾兲 have been demonstrated. 共␪,␾兲共␲ 4, ␲ 6兲共 ␲ 4.9, 0兲共 ␲ 3.7, ␲ 9兲 ␣⌽ 9.01 9.21 8.92 ␥⌽ 6.54 6.44 6.86 ␦noise,⌽ 0.139 0.150 0.169

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Through the method given by Bourennane et al. 关12兴, we

derive the operator W_⌽p共␪,␾兲 and have ␣_⌽p= cos2_共_␪_{兲 for 0}

⬍␪ⱕ␲/ 4 and␣_⌽p= sin2_共_␪_{兲 for}_␲_{/ 4}_ⱕ_␪_⬍_␲_{/ 2. Table}_I

sum-marizes the parameters␥_⌽utilized to prove that the proposed operators are indeed EWs for detecting truly multipartite en-tanglement.

In addition, we are concerned with the robustness to noise for the witnessW_⌽共␪,␾兲. The robustness of W_⌽共␪,␾兲 de-pends on the noise tolerance p_noise⬍␦_noiseis such that

兲典 class.

An approach to derive Cˆ_⌽shown above can be applied to the cases for arbitrary number of qubits straightforwardly. One can formulate sets of correlator operators to identify correlations between two subsystems under two LMSs and then construct the witness operators further. In particular, we have found that the proposed method also provides an ana-lytical and systematic way to construct correlator operators for entangled states with local stabilizers and the correspond-ing EWs as the previous results关13,19兴.

Before proceeding further, let us give a brief summary and conclusion for this section. We have demonstrated a sys-tematical method to derive correlator operators utilized to construct witness operators. The proposed correlator opera-tors are based on necessary conditions of some pure multi-partite entangled state to be created experimentally. More-over, in the example, these witness operators can be measured with only two LMSs. In what follows, we will give two EWs in which the correlator operators can be con-structed systematically. Through these cases for entangle-ment detection, one could realize that the proposed condi-tions of quantum correlacondi-tions possess a wide generality.

II. EWs FOR MULTIPARTITE ENTANGLED STATES A. Detection of genuine multipartite entanglement of the

four-qubit singlet state

Very recently, four-party quantum secret sharing has been demonstrated via the resource of four photon entanglement 关18兴, which is called the four-qubit singlet state 关17兴.

Through the same method presented in the Introduction, we give an EW to detect the four-qubit singlet state.

The four-qubit singlet state is expressed as the following form: 兩⌿典 =

_冑

1 3

冋

兩0011典z+兩1100典z− 1 2共兩0110典z+兩1001典z+兩0101典z +兩1010典z兲

册

. 共28兲

Under the LMS M_4z, we formulate eight sets of criteria for identifying quantum correlation between a specific party and others: the first type of identifications include the following four sets of correlators:

Cˆ_0,m共z兲 = 0ˆ1z0ˆ2z1ˆ3z1ˆ4z− Xm共0ˆ1z0ˆ2z1ˆ3z1ˆ4z兲Xm, 共29兲

Cˆ_1,m共z兲 = 1ˆ1z1ˆ2z0ˆ3z0ˆ4z− Xm共1ˆ1z1ˆ2z0ˆ3z0ˆ4z兲Xm, 共30兲

where Xm=␴x is performed on the mth party for m = 1 , . . . , 4. Then, the second type of criteria are formulated as

Cˆ_0n,k共z兲 =关0ˆ_共2n+1兲z1ˆ_共2n+2兲z− Xk共0ˆ_共2n+1兲z1ˆ_共2n+2兲z兲Xk兴 ⫻共0ˆ_共2n丣3兲z1ˆ共2n丣4兲z+ 1ˆ共2n丣3兲z0ˆ共2n丣4兲z兲, 共31兲

Cˆ_1n,k共z兲 =关1ˆ_共2n+1兲z0ˆ_共2n+2兲z− Xk共1ˆ_共2n+1兲z0ˆ_共2n+2兲z兲Xk兴 ⫻共0ˆ_共2n丣3兲z1ˆ共2n丣4兲z+ 1ˆ共2n丣3兲z0ˆ共2n丣4兲z兲, 共32兲

where k =共2n+1兲,共2n+2兲 for n=0,1; and the symbol “丣” behaves as the addition of modulo 4 for n = 1 and as an ordinary addition for n = 0. The expectation values of the operators Cˆ_l,m共z兲 and Cˆ_ln,k共z兲 for the pure four-qubit singlet state can be evaluated directly and are given by C_l,m,共z兲 _⌿= 1 / 3 and C_ln,k,共z兲 _⌿= 1 / 6 for l = 0 , 1.

It is easy to see that the conditions involved in the expec-tation values of Cˆ_l,m共z兲 and Cˆ_ln,k共z兲 :

C_0,m共z兲C_1,m共z兲 ⬎ 0 and C_0n,k共z兲 C_1n,k共z兲 ⬎ 0, 共33兲 are necessary for the pure four-qubit singlet state. The proof of this statement is similar to the one for proposition 1 pre-sented in the first section.

For invariance of the wave function presented in the eigenbasis of␴x共␴y兲, in analogy, we can construct eight sets of Hermitian operators

共Cˆ0,m关x共y兲兴,Cˆ1,m关x共y兲兴兲 and 共Cˆ0n,k关x共y兲兴,Cˆ1n,k关x共y兲兴兲,

via the replacement of the index z in above Hermitian opera-tors by the index x共y兲 and constructing the operators in the eigenbasis of␴x_共y兲. The expectation values of the above op-TABLE II. Expectation values of three proposed EWs including

W⌽共␲4, ␲ 6兲, W⌽共 ␲ 4.9, 0兲, and W⌽共 ␲ 3.7, ␲

9兲 for the pure states 兩⌽典: 兩⌽共␲4, ␲ 6兲典, 兩⌽共 ␲ 4.9, 0兲典, and 兩⌽共 ␲ 3.7, ␲ 9兲典. 兩⌽典兩⌽共␲4, ␲ 6兲典兩⌽共 ␲ 4.9, 0兲典兩⌽共 ␲ 3.7, ␲ 9兲典 Tr关W_⌽共␲₄,₆␲兲

兩

⌽典具⌽兩兴 −1.45 −1.83 −1.72 Tr关W_⌽共_4.9␲, 0兲

兩

⌽典具⌽兩兴 −1.25 −1.63 −1.52 Tr关W_⌽共_3.7␲,₉␲兲

兩

⌽典具⌽兩兴 −1.55 −1.92 −1.81

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erators are all positive for the state兩⌿典, and so we have the following necessary conditions of the state兩⌿典:

C_0,m关x共y兲兴C_1,m关x共y兲兴⬎ 0 and C_0n,k关x共y兲兴C_1n,k关x共y兲兴⬎ 0. 共34兲 Then, we combine all of the correlator operators proposed above: Cˆ_⌿= Cˆ_⌿共x兲+ Cˆ_⌿共y兲+ Cˆ_⌿共z兲, 共35兲 where Cˆ_⌿共i兲=

兺

l=0 1 共5

兺

m=1 4 Cˆ_l,m共i兲 +

兺

n=0 1

兺

k=2n+1 2n+2 Cˆ_ln,k共i兲 兲共36兲 for i = x , y , z, and present a EW to detect the four-qubit sin-glet state. The following witness operator detects truly mul-tipartite entanglement for states close to the state兩⌿典:

W⌿=␣⌿1 − Cˆ⌿, 共37兲

where␣_⌿= 36.5.

We use the method utilized forW_⌽共␪,␾兲 to prove W_⌿is a EW. First, we seek the witness operatorW_⌿p. Through Ref. 关12兴, the operator is given by:

W_⌿p = 3

41 −兩⌿典具⌿兩. 共38兲

Then, we have to show that if a state ␳ satisfies Tr共W_⌿␳兲 ⬍0, it also satisfies Tr共W_⌿p_␳_{兲⬍0. We find that} _␥

⌿= 30 is

such thatW_⌿−␥_⌿W_⌿pⱖ0.

The sets of correlator operators Cˆ_⌿共x兲, Cˆ_⌿共y兲, and Cˆ_⌿共z兲 note that only three LMSs are used in the witness operatorW_⌿. The number of LMSs is smaller than the required one, 15 LMSs, in Ref.关12兴. Moreover, the robustness of the witness

W_⌿ is specified by ␦noise,⌿= 15/ 88⯝0.170455. This result

satisfies the experimental requirement of robustness in Ref. 关12兴.

B. Detection of genuine multipartite entanglement for a four-level tripartite system

In order to show further utilities of the proposed ap-proach, we proceed to provide a witness to detect genuine multipartite entanglement close to a four-level tripartite GHZ state关14兴: 兩GHZ4⫻3典 = 1 2

兺

_l=0 3 兩l典1z丢兩l典2z丢兩l典3z. 共39兲

First of all, with the knowledge of the wave function repre-sented in the eigenbasis:兩l典nzfor n = 1 , 2 , 3, we have nine sets of correlator operators for identifying quantum correlation between the nth party and others, and are given by

Cˆ_nk,j共z兲 =共kˆ − sˆkj兲nzkˆpzkˆqz 共40兲 for j = 1 , . . . , 9; k = 0 , . . . , 3; n , p , q = 1 , 2 , 3, and n⫽p⫽q; where sˆkj= 0ˆ , . . . , 3ˆ; kˆ⫽sˆkj, and sˆkj⫽sˆk_⬘j for k⫽k

⬘

; and Cˆ_nk,j共z兲 ⫽Cˆ

nk,j⬘

共z兲 _{for j}_{⫽ j}

_⬘

_{. To show C}_ˆ

nk,j

共z兲 _{explicitly, let us take}

the following set of operators numbered by j = 1, for ex-ample,

Cˆ_n0,1共z兲 =共0ˆnz− 1ˆnz兲0ˆpz0ˆqz, Cˆ_n1,1共z兲 =共1ˆnz− 2ˆnz兲1ˆpz1ˆqz, Cˆ_n2,1共z兲 =共2ˆnz− 3ˆnz兲2ˆpz2ˆqz, Cˆn3,1共z兲 =共3ˆnz− 0ˆnz兲3ˆpz3ˆqz.

Another example for the second set of operators j = 2 could be the following one:

Cˆ_n0,2共z兲 =共0ˆnz− 2ˆnz兲0ˆpz0ˆqz, Cˆ_n1,2共z兲 =共1ˆnz− 3ˆnz兲1ˆpz1ˆqz, Cˆ_n2,2共z兲 =共2ˆnz− 0ˆnz兲2ˆpz2ˆqz, Cˆ_n3,2共z兲 =共3ˆnz− 1ˆnz兲3ˆpz3ˆqz.

We progress to a correlation condition for the pure four-level tripartite GHZ state by the following proposition.

Proposition 2. If the expectation values of Cˆ_nk,j共z兲 for some state are all positive for k = 1 , . . . , 3 under some j, the out-comes of measurements for the party n and the rest of the systems are correlated.

Proof. If the nth party is independent of the rest of the system, we can cast the expectation values of the operators Cˆ_nk,j共z兲 as

C_nk,j共z兲 =关P共vn= k兲 − P共vn= skj兲兴P共vp= k,vq= k兲. Since P共vp= k ,vq= k兲ⱖ0, Cnk,j

共z兲 _{should not be all positive.}

Thus Cˆ_nk,j共z兲 ⬎0 for all k’s implies that the measured outcomes for the party n and the rest are correlated. Q.E.D.

All of the expectation values of the operators Cˆ_nk,j共z兲 for the pure four-level tripartite GHZ state are given by C_nk,j,GHZ

4⫻3

共z兲 _{= 1 / 4, which are greater than zero. We then}

con-sider that Cˆ_nk,j共z兲 ⬎0 as a necessary condition of the state. Sec-ond, if an observable with the eigenvector

兩g典nf=1 2

兺

_h=0

3

exp

冋

− i2␲h

4 g

册

兩h典nz 共41兲

for g = 0 , . . . , 3, is measured for each party n = 1 , 2 , 3, we give nine sets of correlator operators to identify quantum correla-tion between the nth party and others

Cˆ_nk,j共f兲 =共kˆ − sˆkj兲nfVˆklr, 共42兲 where Vˆklr=

兺

l,r=0 3 ␦„共k + l + r兲mod 4,0…lˆpfrˆqf 共43兲 and definitions of kˆ, sˆkj, n, p, q, and j are the same as the ones mentioned for Cˆ_nk,j共z兲 . For j = 1, the set of operators speci-fied by the above equations could be:

Cˆn0,1共f兲 =共0ˆ − 1ˆ兲共0ˆ0ˆ + 1ˆ3ˆ + 2ˆ2ˆ + 3ˆ1ˆ兲, Cˆn1,1共f兲 =共1ˆ − 2ˆ兲共0ˆ3ˆ + 1ˆ2ˆ + 2ˆ1ˆ + 3ˆ0ˆ兲, Cˆn2,1共f兲 =共2ˆ − 3ˆ兲共0ˆ2ˆ + 1ˆ1ˆ + 2ˆ0ˆ

+ 3ˆ3ˆ兲, Cˆn3,1共f兲 =共3ˆ − 0ˆ兲共0ˆ1ˆ + 1ˆ0ˆ + 2ˆ3ˆ + 3ˆ2ˆ兲. For j = 2, we could give the set of operators as follows:

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Cˆ_n0,2共f兲 =共0ˆ − 2ˆ兲共0ˆ0ˆ + 1ˆ3ˆ + 2ˆ2ˆ + 3ˆ1ˆ兲, Cˆ_n1,2共f兲 =共1ˆ − 3ˆ兲共0ˆ3ˆ + 1ˆ2ˆ + 2ˆ1ˆ + 3ˆ0ˆ兲, Cˆn2,2共f兲 =共2ˆ − 0ˆ兲共0ˆ2ˆ + 1ˆ1ˆ + 2ˆ0ˆ

+ 3ˆ3ˆ兲, Cˆn3,2共f兲 =共3ˆ − 1ˆ兲共0ˆ1ˆ + 1ˆ0ˆ + 2ˆ3ˆ + 3ˆ2ˆ兲.

Please note that in order to have compact forms, we have omitted the subscripts nf, pf, and qf from the above ex-amples. A correlation condition similar to the one discussed in proposition 2 is proposed by the statement,⬙if the expec-tation values of Cˆ_nk,j共f兲 are all positive for k = 1 , . . . , 3 under some j, there are correlations between the measured out-comes for the party n and the rest of the systems.⬙ Since all of the expectation values of the operators Cˆ_nk,j共f兲 for the pure four-level tripartite GHZ state are greater than zero, i.e., C_nk,j,GHZ

4⫻3

共f兲 _{= 1 / 4, the correlation condition C}_ˆ

nk,j

共f兲 _{⬎0 is then}

necessary for the state.

Therefore, through a linear combination of all of the cor-relator operators proposed above

Cˆ_GHZ 4⫻3=

兺

n=1 3

兺

j=1 9

兺

k=0 3 共1.5Cˆnk,j共z兲 + Cˆ_nk,j共f兲兲, 共44兲 the following witness operator detects genuine multipartite entanglement for states close to兩GHZ_4⫻3典:

WGHZ4⫻3=␣GHZ4⫻31 − CˆGHZ4⫻3, 共45兲

where ␣_GHZ

4⫻3= 40.5. We take an approach similar to the

ones used in the previous proofs for EWs to prove that the above witness operator detects genuine multipartite entangle-ment. In order to show that if an experimental output state␳ satisfies Tr共WGHZ₄_⫻3␳兲⬍0, the state ␳ also satisfies

Tr共WGHZ_4⫻3

p _␳_{兲⬍0, first, we deduce that}

WGHZp ₄_⫻3=

1

41 −兩GHZ4⫻3典具GHZ4⫻3兩, 共46兲 by the method proposed in Ref. 关12兴. Further, through the

relationW_GHZ

4⫻3− 36WGHZ4⫻3

p _{ⱖ0 for the proposed witness}

operator, we then conclude thatWGHZ₄_⫻3can be used to

de-tect truly multipartite entanglement.

Furthermore, when a state mixes with white noise the proposed EW is very robust and it detects genuine multipar-tite entanglement if pnoise⬍0.4. Thus, two local measurement

settings are sufficient to detect genuine four-level tripartite entanglement around a pure four-level tripartite GHZ state.

III. BI FOR ARBITRARY HIGH-DIMENSIONAL BIPARTITE SYSTEMS

In order to derive a BI, we will begin with specifications of correlation conditions for quantum correlation of a two-qudit entangled state. Then, we will proceed to verify that any local hidden variable theory cannot reproduce the corre-lations embedded in the entangled state. This approach is opposite to the one presented in Ref.关9兴.

First, to specify the quantum correlation embedded in the two-qudit entangled state

兩␺d典 =

_冑

1 d

兺

l=0

d−1

兩l典1z丢兩l典2z, 共47兲

we describe the wave function in the following eigenbasis of some observable Vˆ_k共q兲: 兩l典kq=

_冑

1 dm=0

兺

d−1 exp

冋

i2␲m d 共l + nk 共q兲_兲

册

_兩m典kz, _共48兲

for k , q = 1 , 2, where n₁共1兲= 0, n₂共1兲= 1 / 4, n₁共2兲= 1 / 2, and n₂共2兲= −1 / 4 correspond to four different LMSs Mij=共Vˆ₁共i兲, Vˆ₂共j兲兲 for i , j = 1 , 2. From our knowledge of the four different represen-tations of the state 兩␺d典, we give four sets of correlators of quantum correlation Cm共12兲= P关v1共1兲=共− m兲mod d,v2共2兲= m兴 − P关v₁共1兲=共1 − m兲mod d,v₂共2兲= m兴, 共49兲 Cm共21兲= P关v1共2兲=共d − m − 1兲mod d,v2共1兲= m兴 − P关v1共2兲=共− m兲mod d,v2共1兲= m兴, 共50兲 Cm共qq兲= P关v1共q兲=共− m兲mod d,v2共q兲= m兴 − P关v1共q兲=共d − m − 1兲mod d,v2共q兲= m兴共51兲

for m = 0 , 1 , . . . , d − 1 and q = 1 , 2. The superscripts 共ij兲, 共i兲, and共j兲 indicate that some LMS Mij has been selected. For the pure state 兩␺d典 under Mij, the correlator C_m共ij兲 can be evaluated analytically关9兴 and is given by

C_m,_␺ d

共ij兲 ₌ 1

2d3关csc

2_共_␲_{/4d兲 − csc}2_共3_␲_/4d兲兴, _共52兲

where csc共h兲 is the cosecant of h. Since C_m,_␺

d

共ij兲 _{⬎0 for all m’s}

with any finite value of d, we ensure that there are correla-tions between outcomes of measurements performed on the state兩␺d典 under four different LMSs. The proof of this state-ment is similar to that for proposition 2. Hence the correla-tion condicorrela-tions

Cm共ij兲⬎ 0 共53兲

are necessary for the pure two-qudit entangled state兩␺d典. Thus, we take the summation of all C_m共ij兲 ’s

Cd= C共11兲+ C共12兲+ C共21兲+ C共22兲, 共54兲 where C共ij兲=兺_m=0d−1C_m共ij兲, as an identification of the state 兩␺d典. We can evaluate the summation of all C_m共ij兲’s for the state 兩␺d典, and then we have

Cd,␺_d= 2 d2关csc

2_共_␲_{/4d兲 − csc}2_共3_␲_/4d兲兴. _共55兲

One can find that Cd,␺_d is an increasing function of d. For instance, if d = 3, one has C_3,␺

3⯝2.87293. In the limit of

large d, we obtain limd_→⬁Cd,␺_d=共16/3␲兲2⯝2.88202. We proceed to consider the maximum value of Cd for local hidden variable theories which is denoted by Cd,LHV. The following proof is based on deterministic local models

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which are specified by fixing the outcome of all measure-ments. This consideration is general since any probabilistic model can be converted into a deterministic one关20兴.

Sub-stituting a fixed set共v˜₁共1兲,˜v₂共1兲,˜v₁共2兲,˜v₂共2兲兲 into

Cm共ij兲= P共v1共i兲=␣m共ij兲,v共2兲2 = m兲 − P共v1共i兲=␤m共ij兲,v2共j兲= m兲,

where ␣_m共ij兲 and ␤_m共ij兲 denote the values involved in Eqs. 共50兲–共52兲, then we have the result

C_m,LHV共ij兲 =␦共␣共ij兲m ,˜v1共i兲兲␦共m,v˜共j兲2 兲 −␦共␤m共ij兲,˜v1共i兲兲␦共m,v˜2共j兲兲,

共56兲 where␦共x,y兲 denotes the Kronecker delta symbol. Accord-ingly, Cd for local hidden variable theories turns into

Cd,LHV=␦„共v˜1共1兲+v˜共1兲2 兲mod d,0… −␦„− 共v˜1共1兲+˜v2共1兲兲mod d,1…

+␦共v˜₁共1兲+v˜₂共2兲兲mod d,0 −␦_„共v˜₁共1兲+˜v₂共2兲兲mod d,1… +␦„共v˜1共2兲+˜v2共2兲兲mod d,0… −␦„− 共v˜1共2兲

+˜v₂共2兲兲mod d,1… +␦„− 共v˜1共2兲+˜v2共1兲兲mod d,1…

−␦„共v˜1共2兲+˜v2共1兲兲mod d,0…. 共57兲

There are three nonvanishing terms at most among the four positive ␦ functions and there exist four cases for it, for example, one is that if ␦(共v˜₁共1兲+˜v₂共1兲兲mod d,0)=␦(共v˜₁共1兲

+˜v₂共2兲兲mod d,0)=␦(共v˜₁共2兲+˜v₂共2兲兲mod d,0兲=1 is assigned, we obtain ˜v₂共1兲=˜v共2兲₂ and then deduce that ␦(−共v˜₁共2兲

+˜v₂共1兲兲mod d,1)=0. We also know that there must exist one nonvanishing negative␦ function and three vanishing nega-tive ones in the Cd,LHV under the same condition. In the example, the case is ␦(共v˜₁共2兲+˜v₂共1兲兲mod d,0)=1. With these facts, we conclude that Cd,LHVⱕ2. One can check other three cases for the four positive␦functions, and then they always result in the same bound. Thus, we realize that Cd,␺_d ⬎Cd,LHVand the quantum correlations are stronger than the ones predicted by the local hidden variable theories.

For d = 2, the proposed inequality C2,LHVⱕ2 can be

ex-pressed explicitly in the form C ˜共11兲_{+ C}˜共12兲_{+ C}˜共22兲_{− C}˜共21兲_{ⱕ 2,} _共58兲 where C˜共ij兲=兺_k=01 共−1兲k_␦₍_共v˜ 1 共i兲₊_˜_v 2

共j兲_{兲mod d,k), and then we}

obtain the result which is known as the CHSH inequality after the discovery of Clauser, Horne, Shimony, and Holt关8兴.

On the other hand, from the quantum-mechanical point of view, we have a violation of the CHSH inequality by C_2,␺

2

= 2

冑

2.

A surprising feature of the inequality is that the total num-ber of detection events required for analyses by each of the presented correlation functions C共ij兲 is only 2d, which is much smaller than the result O共d2_{兲 shown in Ref. 关}₁₆_{兴. This}

implies that the proposed correlation functions contain only the dominant terms to identify correlations. However, the

proposed BI is nontight from a geometric point of view关21兴.

Since the number of linear independent generators contained in the hyperplane Cd,LHV= 2 is only 4d关19兴 which is smaller than 4d共d−1兲 involved in the condition of tightness 关21兴, the

BI is nontight.

Furthermore, if an experimental output state suffered from white noise and turned into a mixed one with the form

␳=pnoise

d2 1 +共1 − pnoise兲兩␺d典具␺d兩,

the value of Cd for the state ␳ becomes Cd,␳=共1 − pnoise兲Cd,␺_d. If the criterion Cd,␳⬎2, i.e.,

pnoise⬍ 1 −

2 Cd,␺_d

共59兲 is imposed on the system, one ensures that the mixed state still exhibits quantum correlations in outcomes of measure-ments. For instance, to maintain the quantum correlation for the limit of large d, the system must have p_noise⬍0.30604.

On the other hand, it is worth comparing the noise toler-ance of Cdwith the one of the following EW:

W_␺ d

p =1

d1 −兩␺d典具␺d兩. 共60兲 Let the noise fraction be the form pnoise= 1 −⑀, where⑀is a

positive parameter. Then satisfying the condition of entangle-ment Tr共W_␺

d

p _␳_{兲⬍0 implies that} _⑀_{⬎1/共d+1兲. Therefore, in} the case where d→⬁, any state with pnoise⬍1 is detected as

an entangled one. Hence, there is a significant difference between the noise tolerance of Cdand the one ofW_␺

d

p in the limit of large d.

IV. SUMMARY

Through the necessary condition of quantum correlation we develop a systematic approach to derive correlator opera-tors for BIs and EWs. The d-level bipartite BI is strongly resistant to noise and can be tested with fewer analyses of measurement outcomes. The proposed EWs for the general-ized GHZ, four-qubit singlet, and four-level tripartite GHZ states are robust to noise and require fewer experimental ef-forts to be realized. Therefore, the correlation conditions for quantum correlation involved in the approach to construct correlator operators utilized in EWs and BIs can be consid-ered as a connection between them. The generality of the approach widely cover several共different兲 tasks of entangle-ment detections and pave the way for further studies on en-tangled qudits.

We thank J. W. Pan, Z. B. Chen, and Q. Zhang for useful discussions. This work was supported partially by the Na-tional Science Council of Taiwan under Grant No. NSC95-2119-M-009-030.

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