Detecting nonlocality of noisy multipartite states with the CHSH inequality
Rafael Chaves, Antonio Acín, Leandro Aolita, Daniel Cavalcanti
aa r X i v : . [ qu a n t - ph ] M a y Detecting nonlocality of noisy multipartite states with the CHSH inequality
Rafael Chaves, Antonio Ac´ın, Leandro Aolita, and Daniel Cavalcanti Institute for Physics, University of Freiburg, Rheinstrasse 10, D-79104 Freiburg, Germany ICFO-Institut de Ci`encies Fot`oniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain Dahlem Center for Complex Quantum Systems, Freie Universit¨at Berlin, Berlin, Germany
The Clauser-Horne-Shimony-Holt inequality was originally proposed as a Bell inequality to de-tect nonlocality in bipartite systems. However, it can also be used to certify the nonlocality ofmultipartite quantum states. We apply this to study the nonlocality of multipartite Greenberger-Horne-Zeilinger, W and graph states under local decoherence processes. We derive lower boundson the critical local-noise strength tolerated by the states before becoming local. In addition, forthe whole noisy dynamics, we derive lower bounds on the corresponding nonlocal content for thethree classes of states. All the bounds presented can be calculated efficiently and, in some cases,provide significantly tighter estimates than with any other known method. For example, they revealthat N -qubit GHZ states undergoing local dephasing are, for all N , nonlocal throughout all thedephasing dynamics. PACS numbers: 03.67.-a, 03.67.Mn, 42.50.-p
I. INTRODUCTION
Non-locality refers to correlations between the mea-surement results of distant systems that cannot be ex-plained by local hidden-variable (LHV) models [1, 2].The correlations consistent with a LHV model necessar-ily satisfy a set of linear constraints known as
Bell in-equalities [2], which can be experimentally tested. Thus,the violation of any Bell inequality reveals the presenceof non-locality. In addition, apart from a fundamen-tal issue, the detection of nonlocal correlations is alsoof practical relevance. First, the violation of a Bell in-equality is a device-independent entanglement witness,i.e. it allows one to certify entanglement in situationswhere the sources and measurements implemented aretotally unknown [2, 3]. Second, the efficacy at solvinginformation-theoretic tasks such as communication com-plexity problems [4], device-independent quantum keydistribution [5–7] and randomness extraction [8, 9] oramplification [10–12] relies on the presence of nonlocality.Experimentally-friendly ways to extract nonlocal correla-tions from quantum states appears thus highly desirable.The simplest way to do this, in the case of two partswith two dichotomic measurements each, is through theCHSH inequality [13]
CHSH ≡ h a b i + h a b i + h a b i − h a b i ≤ , (1)where a x = ± b y = ± x = { , } and y = { , } for Alice and Bob, respectively, and h a x b y i = p ( a = b | xy ) − p ( a = b | xy ) stands for the statistical average of a x b y . In quantum mechanics these averages can be ex-pressed by h a x b y i ≡ Tr[ ˆ A x ⊗ ˆ B y ρ ], where ˆ A x and ˆ B y are Hermitian observables with eigenvalues ± ρ aquantum state. The CHSH inequality (and its symme-tries) is the only relevant Bell inequality in the bipartitescenario with two dichotomic measurements [14], i.e. itcan tightly capture all non-local correlations. Further-more, for two-qubit quantum states, CHSH violation can be immediately checked via the necessary and sufficientcondition found in [15].In the multipartite scenario, however, the situationchanges drastically. For instance, already for the mod-est case of three parts applying two dichotomic mea-surements each, there are 46 inequivalent classes of non-trivial and tight Bell inequalities [16]. In general, theefficiency in the characterization of nonlocality as thenumber of parts, measurements or outcomes increasesbecomes a major issue. In fact, deciding the compatibil-ity of a given probability distribution with LHV modelsis known to be an NP-complete problem [17, 18].In this paper, we study the nonlocality of gen-uinely multipartite N -qubit Greenberger-Horne-Zeilinger(GHZ), W and graph states under local decoherence pro-cesses described by Pauli channels. We derive lowerbounds on the critical local-noise strength tolerated bythe states before becoming local, in a similar spirit as in[19]. In addition, for each noise strength, we derive lowerbounds on the nonlocal content [20] of the correlationson the three classes of states. The bounds we derive arebased on the CHSH violation of two out of the N qubitsconditioned on a measurement outcome of all other N − N -independent.Furthermore, we show that the estimates given by thesebounds are (in some cases exponentially) tighter thanthose given by any other known method.In Sec. II we introduced the different classes of states,the noise models and figures of merit for nonlocality tobe used in this paper. In Sec. III we describe the generalmethod that is applied in Sec. IV to derive, respectively,lower bounds on the critical noise strength and the non-local content. In V we present a summary of the resultswhile some technical results about graph states Bell in-equalities are relegated to the Appendix. II. STATES, NOISE MODELS AND FIGURESOF MERIT
In this section, we introduce basic notation, define thestates studied, the noise channels considered and the fig-ures of merit we use to assess the non-locality of noisystates.
A. States under scrutiny
We consider three paradigmatic families of genuinelymultipartite N -qubit quantum states: • GHZ states [22] | GHZ N i . = 1 √ | i ⊗ N + | i ⊗ N ); (2) • W states [23] | W N i . = 1 √ N ( | . . . i + | . . . i + . . . + | . . . i ) ;(3) • Graph states [24, 25]. A graph-state | G i is associ-ated to an N -vertex mathematical graph G , whosegeometry is determined by a set E of edges { i, j } indicating which vertices i and j are connected, for1 ≤ i, j ≤ N . More precisely, | G i . = CZ E | + i ⊗ N , (4)being | + i . = ( | i + | i ) / √ CZ E . = Q { i,j }∈E CZ i,j , where CZ i,j . = e ( Z i − i ) ⊗ ( Z j − j ) / ⊗ i,j is the maximallyentangling controlled-Z gate non-trivially actingon qubits i and j , with Z i and Z j the third Paulioperators on qubits i and j , respectively, and i , j , and i,j the identity operators on qubits i , j , and all but i and j , respectively, for any1 ≤ i, j ≤ N . B. Decoherence models
As noise models we consider local Pauli channels of theform Λ( ρ ) . = X i =0 p i σ i ρ σ i . (5)Here, ρ is any initial state and Λ is a single-qubit Paulichannel. σ . =
1, and σ . = X , σ . = Y and σ . = Z referto the usual Pauli operators. The coefficients p i satisfythe relationship p = (1 − p/ p = α p/ p = α p/ p = α p/
2, with α + α + α = 1; so that the total noise strength 0 ≤ p ≤ α , α and α . For exam-ple, the case α = α = 0 describes dephasing along thedirection z of the Bloch sphere (also known as phase-flipchannel). Analogously, α = α = 0 describes dephas-ing along the transversal direction x (bit-flip channel).We consider joint evolutions given by independent andidentical channels on all qubits: ρ p = Λ ⊗ N ( ρ ) . (6) C. Figures of merit
To assess the non-local correlations in quantum states,we focus mainly in two quantities. The first one is thecritical noise strength p c beyond which on no non-localitycan be extracted [19]. We refer to p c as the noise thresh-old and in the following we compute a lower bound toit.The second one is the amount of nonlocality for eachnoise strength p , which we quantify through the EPR2decomposition [20]. Any joint-probability distribution P ,characterising the correlations of some Bell experiment,can be decomposed into convex mixture of a local part P L and a general non-local (no-signalling) part P NL as P = (1 − p NL ) P L + p NL P NL , with 0 ≤ p NL ≤
1. Theminimal non-local weight over all such decompositions,˜ p NL . = min P L ,P NL p NL . (7)defines the nonlocal content of P , which provides a natu-ral quantifyer of the non-locality in P . In turn, we definethe non-local content of a quantum state as the maxi-mum non-local content of correlations over all possibleBell experiments with the state.It turns out that the violation of any Bell inequalityyields a non-trivial lower bound to ˜ p NL [26]. For any(linear) Bell inequality I ≤ I L , with I L the local bound,it is ˜ p NL ≥ I ( P ) − I L I NL − I L , (8)where I NL is the maximum Bell value I over all arbitrarynon-signalling correlations. III. THE METHOD
We will consider a scenario where N parties share amultipartite state and perform local measurements on it.Two of the parties apply two dichotomic measurements,labeled again by x = { , } and y = { , } , with possibleoutcomes a x = ± b y = ±
1, respectively. The other N − N − N − c = ( ± , . . . , ± CHSH c ≡ h a b i c + h a b i c + h a b i c −h a b i c − p ( c ) ≤ , (9)where h a x b y i c = [ p ( a = b | x, y, c ) − p ( a = b | x, y, c )] p ( c ).Notice that this inequality is simply the CHSH inequalitycalculated with the conditional probability distributionfor the two parties given that the other N − c .Proof of the validity of Bell inequality (9): We need toshow that all the local deterministic probability distri-butions, i.e. those assigning definite outcomes for eachmeasurement, satisfy it. For the local deterministic dis-tributions for which p ( c ) = 1, inequality (9) becomes thestandard CHSH inequality (1), while for the local deter-ministic strategies such that p ( c ) = 0 it simply reads0 ≤ (cid:3) Thus, in order to detect nonlocality in a given N -partite state ρ through the inequality (9) we need tofind appropriate local measurements on N − N − N − N − ρ is CHSH = 2 M CHSH ( ρ ) = 2 q t + t , (10)being t and t the two largest eigenvalues of T † T ,with T i,j = tr [( σ i ⊗ σ j ) ρ ], where σ i ⊗ σ j refers to theproduct of the i -th and j -th Pauli operators on the tworemaining qubits, for 1 ≤ i, j ≤
3. So, ρ violates theCHSH inequality if, and only if, M CHSH ( ρ ) > p ( c ) is greater than zero, itsexact value does not affect the critical noise thresholds.Note, however, that using inequality (9) the lower boundfor the nonlocal content will unavoidably depend on p ( c ),namely, ˜ p NL ≥ [CHSH( ρ ) − p ( c )2 . (11)For the states we consider in this paper, p ( c ) will typi-cally decay exponentially with the number of qubits N , also leading to a exponentially decaying lower bound. Inorder to circumvent that and still get non trivial lowerbounds for the nonlocal content we proceed as follows.For all the states we consider (with the exception ofthe W state considered in Sec. IV B) all possible 2 N − measurements outcomes lead to only 2 possible projectedtwo-qubit states that, furthermore, are equivalent up tolocal unitaries. Let us call these projections as events 1and 2, the two respective projected states by ρ and ρ ,and p (1) and p (2) = 1 − p (1) the probabilities of events1 and 2.We can then define Bell inequalities, similar to (9), toevents 1 and 2 as CHSH ≡ h a b i + h a b i + h a b i −h a b i − p (1) ≤ , (12)and CHSH ≡ h a b i + h a b i −h a b i + h a b i − p (2) ≤ , (13)with h a x b y i i = [ p ( a = b | x, y, i ) − p ( a = b | x, y, i )] p i . Fi-nally we use these inequalities to define the following one: CHSH + CHSH ≤ . (14)For most of the states we will consider here, we can findmeasurements A , A , B , B that will lead to p (1) = p (2)and CHSH = CHSH . This, in turn, will imply thatthe lower bound for the nonlocal content will be indepen-dent of the projection probabilities and simply given by˜ p NL ≥ M CHSH ( ρ ) − N − IV. NONLOCALITY THRESHOLD ANDNONLOCAL CONTENT OF NOISY STATES
In this section we show how the multipartite CHSHmethod can be used to calculate the critical noisestrength tolerated by the noisy state before becominglocal. We also compute, for the entire noisy dynam-ics, lower bounds for the nonlocal content of the states.These lower bounds can be significantly better than theones obtained via known multipartite inequalities.
A. Noisy GHZ state
We begin considering GHZ states. In particular, forparallel dephasing, we show that GHZ states of any num-ber of qubits are nonlocal throughout all the noisy dy-namics, a result that cannot be achieved by any otherknown multipartite inequality consisting exclusively offull-correlators.
1. Parallel dephasing
We consider first the detection of nonlocality for theGHZ state (2) undergoing independent dephasing alongthe Z direction. The resulting noisy GHZ state ρ zN canbe expressed as [30, 31] ρ zN = (1 − p ) N | GHZ N i h GHZ N | + (cid:0) − (1 − p ) N (cid:1) ˜ ̺ zN , (15)with ˜ ρ zN = (cid:0) | i h | ⊗ N + | i h | ⊗ N (cid:1) / ZB) in-equalities [32–34]. These encompass all the 2 N tight,linear, full-correlator Bell inequalities in the N -partitescenario where each party makes two dichotomic mea-surements. In particular, of special relevance here is theMermin-Klyshko (MK) inequality [35–37], which is a par-ticular case of the W ZB family. The MK inequality isthe two-setting correlator Bell inequality with the largestviolation in quantum theory [32], with an exponentialmaximal violation 2 ( N − / (the local bound of the MKinequality is given by 1), achieved with the GHZ statefor X and Y measurements.The maximal MK violation for ρ zN can be straightfor-wardly calculated [38] for the case of N odd, to whichwe restrict for simplicity of notation. It is given by2 ( N − / (1 − p ) N and is also attained with X and Y measurements. This yields in turn the noise threshold p zc = 1 − / √ ( N − /N , which is tighter than that givenby any other known multipartite inequality consisting ex-clusively of full-correlators.We next show that the CHSH method renders p zc = 1for all N . Consider local X measurements on the first N − N = 5shows that these measurements are optimal, that is, theymaximize the CHSH violation of the remaining two-qubitstate. See [39] for further details). We consider explicitlythe situation where all N − | + i . However, for anyother outcome the treatment would be equivalent, ex-cept for a local-unitary relabelling of the projected states.This local-unitary equivalence will be explicitly used lateron in order to derive lower bounds to the nonlocal con-tent. The projected two-qubit state conditioned on the N − ρ z = (1 − p ) N | GHZ i h GHZ | + (1 − (1 − p ) N )˜ ρ z . (16)Computing (10) for this state gives M z CHSH = p − p ) N , which is greater than one for all p < N subject to independentparallel dephasing are non-local for any amount of de-phasing p <
1. We stress that such high noise thresholdcannot be detected by any other known multipartite in-equality consisting exclusively of full-correlators.Interestingly, for N = 3 this result can be made evenstronger, since the CHSH method is able to detect thenonlocality in a region where any full-correlator inequal-ity would fail. For the state (15) and N odd it is not difficult to see that only the components of the projectivemeasurement lying in the equatorial plane give a non-nullcontribution for full-correlators. For example for N = 3and the observable O = ( X + Z ) √ ⊗ X ⊗ X we have thattr(O ρ zN ) = tr((X ⊗ X ⊗ X)((1 / √ ρ zN +(1 − / √ p ≥ / N = 3 in the region p ≥ /
2. Transversal dephasing
We now analyse the case of the GHZ state (2) underdephasing along the transversal X direction. The noisystate is now given by ρ xN = X k i =0 , i =1 ,...,N (cid:16) − p (cid:17) N − k p k Π (cid:0)(cid:12)(cid:12) GHZ k N (cid:11) (cid:10) GHZ k N (cid:12)(cid:12)(cid:1) (17)with k = ( k , k . . . k N ), k i = 0 or 1, k = X i =1 ,...,N k i ,and where Π (cid:0)(cid:12)(cid:12) GHZ k N (cid:11) (cid:10) GHZ k N (cid:12)(cid:12)(cid:1) stands for the sum ofall the (cid:0) Nk (cid:1) different permutations of (cid:12)(cid:12) GHZ k N (cid:11) . = X k ⊗ . . . ⊗ X k N N | GHZ N i with X i =
1. The noisy state (17)does not have a simple form as (15), and the optimalmeasurements for the MK inequality depend now on both N and p . Analytical expressions for the MK violationand the corresponding noise threshold as functions of N and p are not available. However, using the multipartiteCHSH method, a straightforward analysis is possible.Applying the projector ( | + i h + | ) ⊗ N − , with supporton all but qubits i and j , to (17) results in the two-qubit state ρ x = (cid:16)(cid:0) − p (cid:1) + (cid:0) p (cid:1) (cid:17) | GHZ i h GHZ | +2 (cid:0) − p (cid:1) (cid:0) p (cid:1) ( X i ⊗ j ) | GHZ i h GHZ | ( X i ⊗ j ). Forthis state, one finds M x CHSH = p − p ) . The noisethreshold obtained is again p xc = 1, independently of N ,which reflects the entanglement robustness of GHZ statesunder transversal local dephasing [42, 43].
3. General Pauli channels
An analytical expression for the GHZ state underthe general Pauli channel (5) can be obtained. Eventhough the evolved state is GHZ-diagonal, analyticalexpressions for the MK violation are again not avail-able. However, the CHSH method offers again a read-ily calculable bound. One obtains then M CHSH = p ( p + p − p − p ) n + ( p − p − p + p ) . As aparticular interesting case, we analyse approximatetransversal local dephasing defined by α = 1 − ǫ , α = ǫ/ α = ǫ/
2. The parameter ǫ thus mea-sures the deviation of perfect transversal dephasing. Inthis case, M CHSH = p (1 − pǫ ) N + (1 − p (1 − ǫ/ ,which, for small values of p , can be approximated as M CHSH ≈ p − p ) Nǫ , yielding an exponential de-cay with N , as with parallel dephasing, but with thedecay rate reduced by a factor ǫ , in a similar fashion towhat happens with the entanglement in these noisy states[42, 43].
4. Non-local content of noisy GHZ states
To obtain a good lower bound for the local contentwe use the inequality (14). For GHZ states (15) underparallel local dephasing, ρ and ρ are given by ρ , = (1 − p ) N (cid:12)(cid:12) GHZ ± (cid:11) (cid:10) GHZ ± (cid:12)(cid:12) + (1 − (1 − p ) N )˜ ρ z . (18)with (cid:12)(cid:12) GHZ ± (cid:11) = (1 √ | i ± | i . In this case p = p = 1 /
2. Choosing A = Z , A = X , B =cos ( θ ) Z + sin ( θ ) X and B = cos ( θ ) Z − sin ( θ ) X wefind the left hand side of (12) and (13) to be equal tocos ( θ ) + sin ( θ )(1 − p ) N . It is a simple calculation toshow that choosing θ = sec − ( p − p ) N ) the lat-ter value equals M z CHSH .So for the GHZ state under parallel dephasing theCHSH method leads to the following lower bound on thenonlocal content˜ p NL ≥ q − p ) N − . (19)In Fig. 1, this bound is compared with the lower boundobtained in Ref. [38] through the MK inequality andwith a numerical estimate, for N = 3. To obtain thenumerical estimate we first note that, for N = 3 and twodichotomic measurements per party, all the facets of lo-cal polytope are known, the so called Sliwa inequalities[16]. We have optimized the violation of Sliwa inequali-ties over all possible projective measurements and using(8) obtained the optimal lower bound on ˜ p NL . As can beseen in Fig. 1, for most of the dynamics, bound (19) istighter than the bound given by the MK inequality.A similar calculation shows that for GHZ states (17)under transversal local dephasing, the CHSH methodgives ˜ p NL ≥ p − p ) − . (20)An analytical expression for the optimal MK violation isnot available, as mentioned before. We numerically op-timise the MK violation and so derive a numerical lowerbound in the nonlocal content, plotted in Fig. 2 togetherwith bound (20). The numerical MK bound is tighter,but the required optimization soon becomes unfeasible as N grows. Bound (20), in contrast, is analytical and doesnot depend on N . CHSH methodMK inequalityNumerical value L o w e r b o und o n FIG. 1. (Color online) Lower bounds on the local content ofGHZ state under parallel dephasing, for N = 3. In red: thebound obtained from the MK inequality [38]; in blue: the newbound (19) from the CHSH method; and in black dashed: thevalue obtained through a numerical optimization described inthe main text. For p > .
18, the nonlocal content is betterdescribed by the bound from the CHSH method.
MK inequality N=3MK inequality N=7MK inequality N=9CHSH method (Size independent) L o w e r b o und o n FIG. 2. (Color online) Lower bounds on the local contentof GHZ state under transversal dephasing. In blue: lowerbound from the MK inequality of N = 3; in red: idem for N = 7; in purple: idem for N = 9; in black dashed: lowerbound (20) from the CHSH method. The bounds from theMK inequality were obtained through numerical optimisationover all possible projective measurements. The CHSH boundis analytical and independent of N . B. Noisy W states
Let us now consider the nonlocality of the noisy Wstate (21). We will consider dephasing along the z di-rection in each of its qubits, which produces the state[44] ρ W N = 1 N (1 − p ′ ) Π ( | . . . i h . . . | ) + p ′ | W i h W | , (21)with p ′ = (1 − p ) and Π( . ) stands for all the permu-tations. The measurement outcome corresponding tothe projector | i h | ⊗ N − (that occurs with probability p = 2 /N ), produces a two-qubit entangled state of theform ρ W for which the CHSH violation is M CHSH = p − p ) . So we recover the result in Ref. [19] thatthe dephased W-state is non-local through all the noisydynamics, that is, p c = 1.Once again we can use the multipartite CHSH methodto provide a lower bound to the nonlocal content of thisstate. However, in this case the project states associatedwith other measurement outcomes other than | i h | ⊗ N − are not local unitarily equivalent to ρ W . Actually theyturn out to be separable and given by | i h | ⊗ N − . Be-cause of that, we must use expression (11) to calculatethe lower bound to the local content, which renders˜ p NL ≥ (2 p − p ) − /N. (22)This bound provides a better estimate for the non-localcontent of (21) when compared to the one that can be ob-tained from the Bell inequality used in Ref. [19]. Therethe inequality used has a non-signalling bound that in-creases exponentially with the number N of qubits, whilethe violation given by the W-state is approximately in-dependent of N . This makes the lower bound decay ex-ponentially while our bound only decays linearly with N . C. Noisy graph-states
As the last application of the multipartite CHSHmethod, we study the nonlocality properties of graphstates (4) subject to Pauli channels. In Ref. [45], mul-tipartite Bell inequalities specially tailored to detect thenonlocality of graph states have been introduced. Forsome of these states, these inequalities are violated ex-ponentially in N . Moreover the violation, for any graphstate under any Pauli channel, can be analytically ex-pressed in a compact closed form (see Appendix).For instance, for graph states under parallel local de-phasing, their violation always decreases exponentiallyfast in N , which implies that the associated lower boundon the local content also decreases exponentially with N .Nevertheless, it is known that the entanglement in graphstates is robust against local noise [46, 47]. With theCHSH method, one easily shows that such entanglementrobustness is also reflected in the non-locality robustness.As an illustration consider a star graph consisting of N − ρ and ρ (with p = p = 1 /
2) are also a two-qubit graph state un-der parallel local dephasing (up to local unitaries). Theleft hand side of (12) and (13) are equal to M CHSH =(1 − p ) √
2. This implies p c = 1 − p / p NL ≥ max [0 , (1 − p ) √ − N and p < p c , this bound is exponentiallytighter than that obtained from the Bell inequalities of[45], as shown in Fig. 3. Graph inequality N=3Graph inequality N=100CHSH method (Size independent) L o w e r b o und o n FIG. 3. (Color online) Lower bounds on the nonlocal contentof for star graph-state of N qubits and | I | = N − N = 3 and N = 100, respectively; in blackdashed: the new bound given by the CHSH method. TheCHSH bound is size independent and offers an exponentiallytighter estimate as N increases. V. SUMMARY
In this work we have used the CHSH inequality in themultipartite scenario, and showed its usefulness to detectthe nonlocality of noisy multipartite states. The methodconsists of locally projecting the multipartite state intoa nonlocal two-qubit state that violates the CHSH in-equality. We have shown examples of states for whichthe nonlocality cannot be detected by the W ZB inequal-ities consisting only of full-correlators (actually, any full-correlator inequality if N = 3), but can be detected bythe present method. The multipartite CHSH methodworks well also in situations were it is difficult to analyt-ically find optimal Bell inequality violations, as for GHZstates undergoing transversal dephasing. Furthermore,the method can be easily applied to obtain tight lowerbounds to the nonlocal content of correlations.We believe these findings should contribute to the de-tection of non-locality for noisy multipartite states. Inparticular, the present method seems to be the simplestone to experimentally detect nonlocality in multipartitestates. ACKNOWLEDGMENTS
We thank N. Brunner for motivating us to write thispaper and the Benasque Center for Science for hospi-tality during the Quantum Information Workshop 2013.We also thank the referee for his/her useful comments.This work was supported by the Excellence Initiative ofthe German Federal and State Governments (Grant ZUK43) the EU project SIQS, and the National Institute ofScience and Technology for Quantum Information, theNational Research Foundation and the Ministry of Edu-cation of Singapore. LA acknowledges the support fromthe EU under Marie Curie IEF No 299141.
Appendix A: Graph states Bell inequalities underPauli channels
Given a vertex i of a graph state | G i of N qubitsand a subset of its neighbors I ⊆ N ( i ), such that noneof the vertices in I are connected by and edge, the Belloperator B ( i, I ) = K i Y jǫI (1 + K j ) ( K i = X i Q j ∈N i Z j arethe generators of the graph, that is, K i | G i = | G i )defines a Bell inequality given by [45] |hBi| = |hB ( i, I ) i| ≤ L ( | I | + 1) , (A1)with a classical bound given by L ( m ) = 2 ( m − / for m odd, and L ( m ) = 2 m/ for m even. The inequality ismaximally violated by the graph | G i with hB ( i, I ) i =2 | I | .Under the action of a Pauli map, the graph statewill turn into a graph diagonal mixed state ρ G = P p µ | G µ i h G µ | , with | G µ i = Z µ ⊗ Z µ · · · ⊗ Z µ N | G i ,where µ = ( µ , . . . , µ N ) is a multi-index, µ j can assumevalues 0 or 1 and the weights p µ depend on the exact form of the Pauli map. The expectation value of the Belloperator B ( i, I ) on this state is given by hBi ρ G = *X p µ K i Y j ∈ I (1 + K j ) | G µ i h G µ | + (A2)= X p µ ( − µ i Y j ∈ I (1 + ( − µ j ) , where we have used that K i | G µ i = ( − µ i | G µ i . From(A2) it follows that the only terms in the convex sum ρ G = P p µ | G µ i h G µ | contributing to the expectationof the Bell operator are µ = (0 , , . . . ,
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