Grothendieck's constant and local models for noisy entangled quantum states

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Grothendieck's constant and local models for noisy entangled quantum states

ACIN, Antonio, GISIN, Nicolas, TONER, Benjamin

Abstract

We relate the nonlocal properties of noisy entangled states to Grothendieck's constant, a mathematical constant appearing in Banach space theory. For two-qubit Werner states ρWp=p∣ψ−⟩⟨ψ−∣+(1−p)1/4, we show that there is a local model for projective measurements if and only if p⩽1/KG(3), where KG(3) is Grothendieck's constant of order 3. Known bounds on KG(3) prove the existence of this model at least for p≲0.66, quite close to the current region of Bell violation, p∼0.71. We generalize this result to arbitrary quantum states.

ACIN, Antonio, GISIN, Nicolas, TONER, Benjamin. Grothendieck's constant and local models for noisy entangled quantum states. Physical Review. A , 2006, vol. 73, no. 6, p. 062105

DOI : 10.1103/PhysRevA.73.062105

Available at:

http://archive-ouverte.unige.ch/unige:36688

Disclaimer: layout of this document may differ from the published version.

1 / 1

Grothendieck’s constant and local models for noisy entangled quantum states

Antonio Acín,¹Nicolas Gisin,² and Benjamin Toner³

1ICFO—Institut de Ciències Fotóniques, Mediterranean Technology Park, 08860 Castelldefels, Barcelona, Spain

2GAP-Optique, University of Geneva, 20, Rue de l’École de Médecine, CH-1211 Geneva 4, Switzerland

3Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA 共Received 23 November 2005; published 9 June 2006兲

We relate the nonlocal properties of noisy entangled states to Grothendieck’s constant, a mathematical constant appearing in Banach space theory. For two-qubit Werner states␳p

W=p兩␺⁻典具␺⁻兩+共1 −p兲1/ 4, we show that there is a local model for projective measurements if and only if p艋1 /K_G共3兲, where K_G共3兲 is Grothendieck’s constant of order 3. Known bounds on K_G共3兲 prove the existence of this model at least for pⱗ0.66, quite close to the current region of Bell violation, p⬃0.71. We generalize this result to arbitrary quantum states.

DOI:10.1103/PhysRevA.73.062105 PACS number共s兲: 03.65.Ud, 03.67.Dd, 03.67.Mn

I. INTRODUCTION

The impossibility of reproducing all correlations observed in composite quantum systems using models similar to that of Einstein, Podolsky, and Rosen共EPR兲 关1兴 was proven in 1964 by Bell. In his seminal work关2兴, Bell showed that all local models satisfy some conditions, the so-called Bell in- equalities, but there are measurements on quantum states that violate a Bell inequality. Therefore, we say that quantum mechanics is nonlocal关3兴. Experimental verification of Bell inequality violation closed the EPR debate, up to some tech- nical loopholes关4兴.

From an operational point of view it is not difficult to define when a quantum state exhibits nonclassical correla- tions. Suppose that two parties Alice共A兲and Bob共B兲share a mixed quantum state␳ with support onHA丢HB, whereHA

共HB兲 is the local Hilbert space of A’s共B’s兲system. Then ␳ contains quantum correlations when its preparation requires a nonlocal quantum resource. Conversely, a quantum state is classically correlated, or separable, when it can be prepared using only local quantum operations and classical communi- cation共LOCC兲. From this definition, due to Werner 关5兴, it follows that a quantum state␳ is separable if it can be ex- pressed as a mixture of product states, ␳⁼兺_i=1^N pi兩␺A

i典具␺A i兩

丢兩␺B i典具␺B

i兩. A state that cannot be written in this form has quantum correlations and is termed entangled. But the above definition, in spite of its clear physical meaning, is somewhat impractical. Tests to distinguish separable from entangled states are complicated关6兴, except whendA= 2 anddB艋3关7兴, dA anddB denoting the dimensions of the local subsystems.

Violation of a Bell inequality by a quantum state is, in many situations, a witness of useful correlations关8兴. In par- ticular, Bell inequality violation is a witness of a quantum state’s entanglement. Now, the question is, are all entangled states nonlocal? For the case of pure states, the answer is yes 关9兴: all entangled pure states violate the Clauser-Horne- Shimony-Holt 共CHSH兲 inequality 关10兴. In 1989, Werner showed that the previous result cannot be generalized to mixed states关11兴. He introduced what are now called Werner states, and gave a local hidden variables 共LHV兲 model for measurement outcomes for some entangled states in this family关5兴. Although the construction only worked for pro-

jective measurements, his result has since been extended to general measurements关12兴.

In spite of these partial results, it is in general extremely difficult to determine whether an entangled state has a local model or not关13兴, since共i兲finding all Bell inequalities is a computationally hard problem关14,15兴and共ii兲the number of possible measurements is unbounded共see, however,关16兴for recent progress兲. This question remains unanswered even in the simplest case of Werner states of two qubits. These are mixtures of the singlet 兩␺⁻典=共兩01典−兩10典兲/

冑

2 with white noise of the form

␳p

W=p兩␺⁻典具␺⁻兩+共1 −p兲1

4. 共1兲

It is known that Werner states are separable if and only if p艋1 / 3, admit a LHV model for all measurements for p艋5 / 12 关12兴, admit a LHV model for projective measure- ments forp艋1 / 2 关5兴, and violate the CHSH inequality for p⬎1 /

冑

2 共see Fig. 1兲. However, the critical value of p, de- notedp_c^W, at which two-qubit Werner states cease to be non- local under projective measurements is unknown. This ques- tion is particularly relevant from an experimental point of view, sincep_c^Wspecifies the amount of noise the singlet tol- erates before losing its nonlocal properties.

In this paper, we exploit the connection between correla- tion Bell inequalities and Grothendieck’s constant关17兴, first

FIG. 1. Nonlocal properties of two-qubit Werner states ␳p W. Werner’s local model works up top= 1 / 2, while the CHSH inequal- ity is violated whenp⬎2^−1/2⬃0.71. Here, we prove the existence of a local model for projective measurements whenpⱗ0.66.

1050-2947/2006/73共6兲/062105共5兲 062105-1 ©2006 The American Physical Society

noticed by Tsirelson关18兴, to prove the existence of a local model for several noisy entangled states. We first demon- strate that p_c^Wis related to a generalization of this constant, namely, p_c^W= 1 /K_G共3兲, where K_G共3兲 is Grothendieck’s con- stant of order 3关19兴. The exact value ofK_G共3兲is unknown, but known bounds establish that 0.6595艋p_c^W艋1 /

冑

2. Thus, we close more than three-quarters of the gap between Wern- er’s result and the known region of Bell inequality violation 共see Fig. 1兲. Next, we show that if Alice共or Bob兲is restricted to make measurements in a plane of the Poincaré sphere, then there is an explicit LHV model for all p艋1 /KG共2兲

= 1 /

冑

2. This improves on the bound of Larsson, who constructed a LHV model for planar measurements for p 艋2 /␲关20兴. Thus, in the case of planar projective measure- ments, violation of the CHSH inequality completely charac- terizes the nonlocality of two-qubit Werner states.

In the case oftraceless two-outcomeobservables, we can extend our results to mixtures of an arbitrary state ␳ ^on C^d丢C^dwith the identity, of the form关21兴

␳p=p␳⁺共1 −p兲1

d². 共2兲

Denote by pc共␳兲 the maximum value of p for which there exists a LHV model for the joint correlation of traceless two- outcome observables on␳p, and define

p_c^d= min

␳ pc共␳兲, pc= lim

d→⬁p_c^d. 共3兲

Thenpc= 1 /KGwhereKGis Grothendieck’s constant. Again, the exact value ofK_Gis unknown, but known bounds imply 0.5611艋p_c艋0.5963.

Finally, we discuss the opposite question of finding Bell inequalities better than the CHSH inequality at detecting the nonlocality of␳p

W, or, more generally, of Bell diagonal states 关22兴. In particular, we show that none of theI_nn22 Bell in- equalities introduced in Ref. 关23兴 is better than the CHSH inequality for these states.

Before proving our results, we require some notation. We write a two-outcome measurement by Alice 共Bob兲 as 兵A⁺,A⁻其 共兵B⁺,B⁻其兲, where the projectors A^± correspond to measurement outcomes ±1. We define theobservablecorre- sponding to Alice’s 共Bob’s兲 measurement as A=A⁺−A⁻ 共B=B⁺−B⁻兲. An observable A is traceless if trA= 0, or equivalently trA⁻= trA⁺. The joint correlation of Alice and Bob’s measurement results, denoted␣^and␤, respectively, is 具␣␤典= tr共A丢B␳兲. 共4兲 Alice’s local marginal is specified by 具␣典= tr共A丢1␳兲, and similarly for Bob. Together,具␣␤典,具␣典, and具␤典define the full probability distribution for two-outcome measurements on␳^. A LHV model for the full probability distribution is one that gives the same values具␣␤典,具␣典, and具␤典as quantum theory.

A LHV model for the joint correlation is one that gives the same joint correlation具␣␤典, but not necessarily the correct marginals. In the qubit case, the projective measurements applied by the parties are specified by the direction of their Stern-Gerlach apparatuses, given by normalized three- dimensional real vectorsaជ ^andbជ_:A=aជ^·␴ជ ^andB=bជ_·␴ជ^.

II. WERNER STATES

Let us first consider the case of Werner states 共1兲. For projective measurements on␳p

W, LHV simulation of the joint correlation is sufficient to reproduce the full probability dis- tribution. This follows from the lemma given below.

Lemma 1. Suppose that there is a LHV modelLthat gives joint correlation 具␣␤典L. Then there is a LHV model L⬘ with the same joint correlation and uniform marginals:

具␣␤典L⬘=具␣␤典L,具␣典L⬘=具␤典L⬘= 0.

Proof. Let␣^and␤be the outputs generated by the LHV L 共dependent on the hidden variables and measurement choices兲. Define a new LHVL⬘ by augmenting the hidden variables ofL with an additional random bitc苸兵−1 , 1其. In L⬘, Alice outputsc␣ ^{and Bobc}␤^. 䊏 Therefore, the analysis of the nonlocal properties of Werner states under projective measurements can be re- stricted to Bell inequalities involving only the joint correla- tion. Actually, this holds for any Bell diagonal state, under projective measurements, since trA␳^{= tr}B␳⁼1/ 2 for all these states, so all projective measurements give uniform margin- als. In the Bell scenarios we consider, Alice and Bob each choose from m observables, specified by 兵A1, . . . ,Am其 and 兵B1, . . . ,Bm其. We can write a generic correlation Bell inequal- ity as

冏

^i,j=1

^兺

^{m Mij具␣i␤j典}

冏

^{艋1, 共5兲}

whereM=共M_ij兲 is anm⫻m matrix of real coefficients de- fining the Bell inequality. The matrixM is normalized such that the local bound is achieved by a deterministic local model, i.e.,

max

a_i=±1,b_j=±1

冏

^i,j=1

^兺

^{m Mijaibj}

冏

^{= 1. 共6兲}

For the singlet state,具␣i␤j典_␺⁻= −aជi·bជ_j. We obtain the maxi- mum ratio of Bell inequality violation for the singlet state, denotedQ, by maximizing over normalized Bell inequalities, and taking the limit as the number of settings goes to infinity:

Q= lim

m→⬁sup

M_ij max

aជi,bជ

j

冏

^i,j=1

^兺

^{m Mijaជi·bជj}

冏

^{. 共7兲}

Since all joint correlations vanish for the maximally mixed state, it follows that the critical point at which two-qubit Werner states do not violate any Bell inequality isp_c^W= 1 /Q.

As first noticed by Tsirelson, the previous formulation of the Bell inequality problem is closely related to the definition of Grothendieck’s inequality and Grothendieck’s constant, K_G 共see 关18兴 for details兲. Grothendieck’s inequality first arose in Banach space theory, particularly in the theory of p-summing operators关24兴. We shall need a refinement of his constant, which can be defined as follows关17兴:

Definition 1. For any integer n艌2, Grothendieck’s con- stant of ordern, denotedKG共n兲, is the smallest number with the following property: Let M be any m⫻m matrix for which

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冏

^i,j=1

^兺

^{m Mijaibj}

冏

^{艋1, 共8兲}

for all real numbersa₁, . . . ,am,b₁, . . . ,bm苸关−1 , + 1兴. Then

冏

^i,j=1

^兺

^{m Mijaជi·bជj}

冏

^{艋KG共n兲 共9兲}

for all unit vectorsaជ1, . . . ,aជm,bជ_{1, . . . ,b}ជ_minRⁿ_. Definition 2. Grothendieck’s constant is defined as

K_G= lim

n→⬁K_G共n兲. 共10兲

The best bounds currently known for K_G are 1.6770艋KG艋␲^/关2 ln共1 +

冑

2兲兴= 1.7822 关25兴. The lower bound is due to Reeds and, independently, Davies关26兴, while the upper bound is due to Krivine关19兴.

It follows immediately from the first definition that the maximal Bell violation for the singlet state共7兲isKG共3兲. We have therefore proved the following theorem.

Theorem 1. There is a LHV model for projective measurements on the Werner state ␳p

W if and only if p艋p_c^W= 1 /KG共3兲.

It is known that

冑

2艋K_G共3兲艋1.5163. The lower bound follows from the CHSH inequality; the upper bound is again due to Krivine关19兴. He shows thatKG共3兲艋␲^/共2c3兲wherec₃ is the unique solution of

冑

c₃ 2

冕

0

c₃

t^−3/2sint dt= 1 共11兲

in the interval 关0 ,␲/ 2兴. Numerically we find that c₃⬇1.0360. This implies KG共3兲艋1.5163 and p_c^W艌0.6595.

Furthermore, it turns out that an explicit LHV model emerges from Krivine’s upper bound onKG共3兲, and the de- tails are presented in关27兴.

Another result follows from Krivine’s work.

Theorem 2. If Alice’s projective measurements are re- stricted to a plane in the Poincaré sphere, then there is a LHV model for␳p

Wif and only if p艋1 /

冑

2.

Proof. In this case, the vectors aជi in Eq.共7兲 are two di- mensional. Since the quantum correlation depends only on the projection ofbជ_jontoaជi, we can assume that the vectorsbជ_j lie in the same plane. It follows thatp_c^W= 1 /KG共2兲for planar measurements, and Krivine has shown thatKG共2兲is equal to

冑

2 关19兴. 䊏

Again Krivine’s proof can be adapted to give an explicit LHV model for planar measurements, valid for p艋1 /

冑

2 关27兴.

III. GENERALIZATION TO HIGHER DIMENSION It is possible to extend these results to general states of the form共2兲, if we restrict our analysis to correlation Bell inequalities of traceless two-outcome observables. Admit- tedly, this analysis is far from sufficient. Indeed it does not

allow us to determine whether the full probability distribu- tion admits a LHV model even in the case of two-outcome measurements, since the most general Bell inequalities have terms that depend on marginal probabilities关23兴. Mindful of this caveat, we now prove the existence of LHV models for the joint correlation of the states共2兲. To make the connection with Grothendieck’s constant, we start with a representation of quantum correlations as dot products, first noted by Tsire- lson 关18兴. It is sufficient to restrict attention to the case of pure states, since we can obtain a LHV for a mixed state␳^by decomposing it into a convex sum of pure states, and taking a convex combination of the LHV’s for those pure states.

Lemma 2. Suppose Alice and Bob measure observablesA andB on a pure quantum state兩␺典苸C^d丢C^d. Then we can associate a real unit vectoraជ苸R^2d2 withA 共independent of B兲, and a real unit vectorbជ_苸R^2d2_withB共independent ofA兲 such that 具␣␤典_␺=aជ^·bជ. Moreover, if 兩␺典 is maximally en- tangled, then we can assume the vectorsaជandbជ_{lie inR}^d2−1_. Proof. Let 兩a典=A丢1B兩␺典 and 兩b典=1A丢B兩␺典. Then 具␣␤典=具a兩b典,具a兩a典=具b兩b典= 1. Denote the components of兩a典 as a_i where i= 1 , 2 , . . . ,d², and similarly for 兩b典. We now define a 2d²-dimensional real vector aជ

=共Rea₁, Ima₁, Rea₂, Ima₂, . . . , Read², Imad²兲, and simi- larlybជ_{=共Reb1, Imb1, Reb2, Imb2}, . . . , Rebd², Imbd²兲. Then aជ·aជ=bជ_·bជ_{= 1 and 具}␣␤典=aជ·bជ _{共because具a兩b典is real兲.}

If 兩␺典 is maximally entangled, we can assume 兩␺典=兩␺⁺典

=共1 /

冑

d兲兺_i=1^d 兩ii典. We calculate具␣␤典_␺⁺= tr_A共AB^t兲/dwhereB^tis the transpose ofB. Introduce a 共d²− 1兲-dimensional basisg_i for traceless operators on HA, normalized such that tr共gigj兲=d␦ij. LetA=兺iaigi,B^t=兺ibigi, which define the vec- torsaជ andbជ. Squaring these definitions and taking the trace gives兺ia_i²=兺ib_i²= 1. Finally, tr共AB^t兲=d兺iaibj, which implies that具␣␤典=兺iaibi=aជ·bជ_.

The converse of Lemma 2 is also true: all dot products of normalized vectors aជ,bជ_苸Rⁿ are realized as observables on 兩␺⁺典, wheren= 2log₂d+ 1 andxdenotes the largest integer less than or equal tox. This result was derived by Tsirelson in Ref.关18兴. For the sake of completeness, we state it here without proof共see关18兴for the details兲.

Theorem 3. Let兵aˆi其_i=1^m and兵bˆj其^m_j=1be sets of unit vectors in Rⁿ. Letd= 2^n/2 and兩␺⁺典be a maximally entangled state on C^d丢C^d. Then there are observables A₁. . . ,A_mandB₁. . . ,B_m onC^dsuch that

具␣i典=具␺⁺兩Ai丢1兩␺⁺典= 0, 共12兲 具␤j典=具␺⁺兩1丢B_j兩␺⁺典= 0, 共13兲

具␣i␤j典=具␺⁺兩Ai丢Bj兩␺⁺典=aˆi·bˆj, 共14兲

for all 1艋i,j艋m.

Note that in our case, the stipulation that the observables be traceless ensures that their outcomes are random on the

maximally mixed state. The next theorem follows from Lemma 2 and Theorem 3.

Theorem 4. Let␳ be a state onC^d丢C^dand define␳pand p_c^das in Eqs.共2兲and共3兲. Then

1

KG共2d²兲艋p_c^d艋 1

KG共2log2d+ 1兲. 共15兲

In other words, there is always a LHV model for the joint correlation of traceless two-outcome observables on ␳p for p艋1 /K_G共2d²兲 and there is a state 共in fact, the maximally entangled state onlog₂dqubits兲such that the joint correla- tion is nonlocal forp⬎1 /K_G共2log₂d+ 1兲.

Corollary 1. The threshold noise for the joint correlation of two-outcome traceless observables ispc= 1 /KG.

This follows from the previous theorem, taking the limit d→⬁. The known bounds imply 0.5611艋pc艋0.5963. Com- pare this to ps, the threshold noise at which the state␳p is guaranteed separable: whileps decreases with dimension at least as 1 /共1 +d兲 关28兴,pcapproaches a constant. In the case of two-qubit systems, we can be more specific, because pro- jective measurements are traceless and have two outcomes.

Corollary 2. Suppose ␳ is an arbitrary state on C²丢C². Then there is a LHV model for the joint correlation on

␳p=p␳⁺共1 −p兲1/ 4 for p艋1 /KG共8兲. In particular, KG共8兲 艋1.6641 关19,27兴, which implies there is a LHV model for p艋0.6009.

For maximally entangled states, marginals of traceless ob- servables are uniform, so Lemmas 1 and 2 imply the follow- ing.

Theorem 5. Let␳p=p兩␺⁺典具␺⁺兩+共1 −p兲1/d²where兩␺⁺典is a maximally entangled state inC^d丢C^d. Then there is a LHV for the full probability distribution arising from traceless ob- servables forp艋1 /K_G共d²− 1兲.

IV. BELL INEQUALITIES FOR WERNER STATES

Just as upper bounds onKG共n兲 yield LHV models, lower bounds yield Bell inequalities. The case of Werner states appears of particular interest: at present, there is no Bell inequality better than the CHSH inequality at detecting the nonlocality of ␳p

W 关29兴. This and other approaches to con- struct new Bell inequalities will be presented in关27兴. Unfor- tunately, none of these inequalities could be proven to be better than the CHSH inequality. It is remarkable how diffi- cult it is to enlarge this region of Bell violation or, equiva- lently, to show thatKG共3兲⬎KG共2兲=

冑

2. Actually, in the case of random marginal probabilities, as for Bell diagonal states under projective measurements, no improvement over the CHSH inequality can be obtained using 3⫻nmeasurements 关30兴.

A similar result can also be proven for the whole family of the so-calledI_nn22 关23兴Bell inequalities. These are speci- fied by a matrix of zeros and ±1 as follows:

I_nn22=

冢

^{−−−共n − 1兲共n − 2兲共n − 3兲− 1]0 − 1]11111 − 1¯ ¯ ¯ ¯¯ ¯ ¯¯ ¯0]1 − 1¯ ¯ ¯]0 ¯ ¯1]0 − 1¯1] − 1]01000}

冣

_共16兲

All the coefficients in the first column共row兲refer to Alice’s 共Bob’s兲marginal probabilities, while the rest of the terms are for joint probabilities. Only one of the two possible out- comes, say +1, appears in the inequality and its local bound is always zero. For example, when n= 2, and denoting p共ai,bj兲=p共ai= + 1 ,bj= + 1兲,I₂₂₂₂reads

p共a1,b₁兲+p共a1,b₂兲+p共a2,b₁兲−p共a2,b₂兲−p共a1= + 1兲

−p共b1= + 1兲艋0, 共17兲

which is equivalent to the CHSH inequality.

Theorem 6. Consider the set ofI_nn22Bell inequalities, for n two-outcome settings. Then, if a Bell diagonal state vio- lates any of these inequalities with projective measurements, it also violates the CHSH inequality.

Proof. Our proof takes advantage of the fact that all mar- ginal probabilities for projective measurements on Bell diag- onal states are fully random. Thus, when dealing with these states, one can put all the terms in the first row and column of共16兲equal to 1 / 2. In order to avoid confusion, we denote byI_n⬘ ^theInn22 inequalities where the local terms have been replaced by 1 / 2.

We start our proof with the simplest nontrivial caseI₃₃₂₂. For Bell diagonal states, it can be written as

I₃⬘⁼¹

2共I₂⬘共1213兲+I₂⬘共1223兲+I₂⬘共1312兲+I₂⬘共2312兲兲艋0, 共18兲 where the arguments of I₂⬘共ijkl兲 are the measurements that appear in theI₂⬘ inequality,iandj for Alice, andkandlfor Bob. From this identity we have that the violation ofI₃⬘ ^im- plies that at least one of the I₂⬘ inequalities is violated too.

This procedure can be generalized for all n: the idea is to expressI_n⬘in terms ofI₂⬘inequalities using the joint probabil- ity terms with a negative sign in 共16兲. For example, when n= 4 one has

I₄⬘⁼¹

3

冉

^{I2⬘共1214兲+I2⬘共1224兲+I2⬘共1234兲+I2⬘共1313兲+I2⬘共1323兲}

+I₂⬘共2313兲+I₂⬘共2323兲+I₂⬘共1412兲+I₂⬘共2412兲+I₂⬘共3412兲 +p共a3,b₃兲−1

2

冊

^{艋0. 共19兲}

Note that since all local probabilities are equal to 1 / 2, p共a3,b₃兲− 1 / 2 is never positive. Thus, whenever I₄⬘⬎0, at least one of theI₂⬘ inequalities appearing in共19兲is violated.

For arbitraryn,I_n⬘can always be written as

ACÍN, GISIN, AND TONER PHYSICAL REVIEW A73, 062105共2006兲

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I_n⬘^{= 1}

n− 1

冋

^s

^兺

^{i=11共n兲I2⬘+s}

^兺

^i=12共n兲

^冉

^{p共a,b兲−12}

^冊 册

^{艋0, 共20兲}

i.e., the sum of s₁共n兲 I₂⬘ inequalities and s₂共n兲 negative terms p共ai,bj兲− 1 / 2, up to an n− 1 factor. Some patient calculation shows that s₁共n兲=n共n²− 1兲/ 6 and s₂共n兲

=共n− 1兲共n− 2兲共n− 3兲/ 6. Thus, if a Bell diagonal state violates I_nn22, it also violates a CHSH inequality. Consequently, none of these inequalities enlarge the known region of Bell viola- tion for Werner states.

After seeing these results, one would be tempted to con- jecture that the CHSH violation provides a necessary and sufficient condition for detecting the nonlocality of Bell di- agonal states, and in particular of Werner states. This result, however, would imply thatK_G共3兲=K_G共2兲=

冑

2, which seems unlikely. Actually, one can find in关25兴an explicit construc- tion with 20 settings showing thatKG共5兲艌10/ 7⬎

冑

2. More recently, one of us has shown thatK_G共4兲⬎

冑

2 as well关27兴.

V. CONCLUSIONS

In this work, we have exploited the connection between Bell correlation inequalities and Grothendieck’s constants to prove the existence of LHV models for several noisy en- tangled states. In the case of Werner states, one can demon- strate the existence of a local model for projective measure- ments up to p⬃0.66, close to the known region of Bell violation. Although we only proved here the existence of the LHV models, the correspondence between noise thresholds and Grothendieck’s constants can also be exploited to con- struct the explicit models. Indeed, these can be extracted from共the proofs of兲 Krivine’s upper bounds onK_G共n兲. The details are presented in Ref.关27兴.

ACKNOWLEDGMENTS

This work is supported by the National Science Founda- tion under Grant No. EIA-0086038, a Spanish MCyT

“Ramón y Cajal” grant, the Generalitat de Catalunya, the Swiss NCCR “Quantum Photonics,” and OFES within the European project RESQ 共Grant No. IST-2001-37559兲. We thank Steven Finch for providing us with Ref.关26兴.

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冑

2 and兩␺^±典=共兩01典±兩10典兲/

冑

2. Werner states are Bell diagonal.

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