Generic quantum nonlocality

Generic quantum nonlocality

PhysicsLettersA 166 (1992) 293—297 North-Holland PHYSICS LETTERS A Generic quantum nonlocality Sandu Popescu School of Physics and Astronomy, Beverl...

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PhysicsLettersA 166 (1992) 293—297 North-Holland


Generic quantum nonlocality Sandu Popescu School of Physics and Astronomy, Beverly and Raymond Sackler Faculty of Exact Sciences, TelAviv University, Ramat Aviv, TelAviv 69978, Israel

and Daniel Rohrlich’ Racah Institute of Physics, The Hebrew University ofJerusalem, Jerusalem 91904, Israel Received 18 February 1992; revised manuscriptreceived 4 May 1992; accepted for publication 12 May 1992 Communicated by J.P. Vigier

For any entangled state of two or more systems, quantum theory predicts experimental results that are inconsistent with local realism: they violate a generalized Bell inequality. But some mixtures of entangled states do not violate any generalized Bell inequality.

When Bell published his celebrated inequalities [1], he established that quantum theory is incompatible with local realism. There are long-range quantum correlations that a theory of local hidden variables can never explain. No choice of values for the hidden variables can reproduce certain quantum predictions unless distant regions can communicate instantaneously. Bell established this incompatibility by analyzing the special case of two spin-i partides coupled in an angular momentum singlet state. For that state he prescribed measurements for which the quantum predictions violate the inequalities. Bell’s special case is a concrete counterexample to the claim that a theory of local hidden variables can reproduce quantum predictions. A natural question is whether this contradiction between quantum predictions and local realism is typical, or whether it is restricted to some very special cases. Few papers have addressed this question. Capasso, Fortunato and Sellen [2] and Gisin [3] showed that any entangled state of two spin-I particles violates a Bell inequality. In a recent paper of Gisin and Peres [4] this resuit extends to pairs of systems with effective Hilbert Supported by the US—Israel Binational Science Foundation.

spaces of arbitrary dimension [5]. In this Letter we extend the result further to any number of systems of any kind. We derive generalized Bell inequalities which hold in any theory of local hidden variables. For any entangled state of two or more systems, we prescribe experiments for which quantum predictions violate these Bell inequalities. Thus, any entangled quantum state leads to a contradiction with local realism. We define what we mean by an entangled state: it is a state involving two or more systems, macroscopically separated, that cannot be factored into a product of states for the individual systems. We first review the results for pairs of particles and then present our extension. Suppose we have an ensemble of particle pairs. We measure eitherA or A’ on one partide in each pair and either B or B’ on the other, where A, A’, B and B’ denote any physical variables with maximum absolute value 1. Let E(A, B), etc., denote the expectation values ofthe products AB, etc., measured on pairs. The Clauser, Home, Shimony and Holt [61 (CHSH) inequality —2 ~ E(A, B) +E(A, B’) +E(A’, B) E(A’, B’) ~ 2

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(1) 293

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holds in any theory of local hidden variables. Considen first the special case of two spin-i particles in an entangled state. An entangled state ~P of two spinI particles can always be written (by choosing an appropriate basis for each particle) as a sum of two terms




Here 1> and I J~>represent spins polarized along axes which may differ from one particle to the other 2+ /32=1 As and may chosen real, withstate. a longaasand a ~/30~ /3, be W is an entangled Since we are considering spin-i particles, the observables could be spin projections onto unit vectors a, a’, S and 6’. We now show how to choose this set ofunit vectors such that the quanta! expectation values for the operators A=o~ A’=~ 1 B=,2 Sand B’ a2 ‘6’ the in the W violate the CHSH inequality.= Thus statestate W leads to nonlocal correlations. The quantum prediction for the sum of expectation values (1) 15

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<°~ ~ = dOS 0 dOS X +2aflsinOsinxcos(~+)



Again, from the form of <~~~y2 é> we know that it takes its maximum with respect to 0 at the value 2


[cosX+4a and thus





/3 sin xcos (~+~)]



max = (cos2x’+4a2fl2 sin2~’)1/2





Thus 2x+ cos2x’ max F( W) = 2 [cos +4 a 202( 2 2 ‘\i1/2 P sin x-~-sin x , We obtain a relation between thogonality of é and ‘



x and x’ from the or-


(3) Theorem 1. For any entangled state, F( W) violates the CHSH inequality for some choice of axes a, a’, S and 6’. The maximum value of F( ~P) i~ 2(1 +4a2fl2)”2. +


To prove theorem 1, we rewrite F( ~P)in terms of unit vectors ê and ê’ such that 6+6’ = 2 cos0 ê and 5—6’ = 2 sin 9 ~ F( ¶1’) = 2
1. ac,2. ê> dos 0 sinO.


(Note that ê~’= 0.) The right-hand-side of eq. (4) has the form Xcos Ysin 2 + Y20+ ) “a. Let0, which has the maximum value (X á= (sin 0 dos ô, sin 0 sin ö, dos 0) and ê= (sin x cos e, sin x sin c, cos x) and likewise for a’ and é’ with primes on angles. We obtain 294

sin~sinX’cos(e—e’)+cos~cos~’=O. 2x’ and substituting we find that = Solving for F( cosW); then dos2x’ = sin2x and we obtain maximizes F( ¶P) = 2(1 + 4a2$2 ) 1/2, as claimed. A set of unit vectors that realizes this maximum value is á=z, â’=~,6=jcos 2t+~sin2t and S’=~cos2t—~sin21, where tan 2t~2afl. This set of unit vectors depends explicitly on !P through a and /3. The dependence is not incidental. For any fixed set of unit vectors, a, a’, Sand 5’, there are entangled states Wwith small enough a on /3 for which the quantum correlations of !P do satisfy eq. (1) [7]. Theorem 1 refers to two-level systems only. The next step is a generalization to two systems in an entangled isstate, whereSuch the anumber of always levels for each system arbitrary. state can be written in block-diagonal form,

~P=aIt>I1>+flI~>I~>+... (10) where 1> and ~>now represent any two states (not necessanly spin) and the dots represent terms orthogonal to the first two. ~Pis entangled if a ~ 0 ~ /3, but we no longer have 2+ p21, which was nec,

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essary for the previous result. But define A, A’, B and B’ such that

A=a1a, A’=a1~a’, B=a2.6 and

on the subspace spanned by I ~> and I ~> and 1 on the orthogonal complement. Since these operators do not connect the states I t> and I J.> with the orthogonal complement, their expectation values are easily computed: they are just weighted averages of the expectation values in each of the subspaces. The weighting is given by the probability to detect a pair in each subspace, which is 22 + + /32) /32 in inthe thesuborspace of I C> and I J.> and 1 (a thogonal complement. In the orthogonal complement the expectation value of all operator products is 1; in the subspace of IC> and I ~,> expectation val—

ues are computed for the state a’IC> II> +fl’l~>I ~>, where a’ a (a2 + /32)_I /2 and /3’ fl( a2 + /32) 1/2 Putting these facts together, we obtain for the sum of quantum expectation values —


<~PlABIW> + < ~PIAB’I w>

+ < ~PIA ‘B I W>

< !P~A ‘B’ I W>

=(a2+fl2)x2[l+4(a’/3’)2] + 2 [1

(a2 + /12)]




which is larger than 2 for a/3~t0, so F( !P) violates the CHSH inequality. We have proved Theorem 2. For any entangled state of two systems, a set of measurements may be prescribed for which the predictions of quantum mechanics are inconsistent with local realism. Although any entangled state of two systems violates the CHSH inequality, the same cannot be said for mixtures of entangled states. (Note that a mixtune of entangled states can arise from a pure entangled state if some of the systems escape detection.) Some mixtures of entangled states do not violate any generalized Bell inequality. The following example suffices: Consider two spin-I particles in a mixture of entangled states (It>1It)~2+I1>1I~>2)/~/’~ and (Ii>~I 1>2 I 1>~I ~>2) i,,/~with equal probability. This mixture is completely equivalent to a mixture of the states I t>~11>2 and I ~ I ~>2 with equal probability, which clearly does not violate any Bell inequality. The two mixtures are equivalent because

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they can arise from the same entangled three-particle state



+ (It>1 I t >2

I ~>l I ~>2)1 ~ >3]

=iE It>~It>2( It>3±I~>3) + I ~> I ~>2(

It>3 I~>3)]


in whichmixture the third goes undetected. Thus neither can particle lead to nonlocal correlations. We would like to extend theorem 2 to an arbitrary number of systems. A special case is due to Greenbergen, Home, Shimony and Zeilinger [8]. The generalized Bell inequality for an entangled state of n systems must involve local measurements on each of the systems; we have just seen that if some systems escape detection, it may be that no Bell inequality is violated. Nor can we, in general, treat some of the n systems as a single system, when they are macroscopically separated. We cannot assume that all nonlocal operators can be measured, since this assumption implies observable violations of the special theory of relativity [9]. However, suppose we have an entangled threeparticle state, ~P.Let us consider correlations between two of the systems, on the assumption that a particular outcome has been obtained from a measurement on the third. A theory of local hidden variables assumes that the result of each measurement depends only on a set of variables A, distributed adcording to a measure p(A) dA, where fp(A) dA = 1. Let the probability that a measurement of A, B and C yields a, b 1 and Ck respectively be denoted P(a1, b~, Ck, A), where the presumed dependence on the A is made explicit. This probability is, by assumption, equal to a product of probabilities P(a,, A )P(b~,A) X P( Ck, A). The correlation between A and B, given that a measurement of C yields Ck, is defined to be

E(A, B, Ck)~ ~

P(a1, b3, Ck, A)p(A) cIA

5 P(a1, b~,Ck, A)p(A)



where, as before, the values a,, b~.are assumed to have maximum norm 1. Since the probabilities factorize, the denominator is simply P(ck), the probability of Ck, and the numerator is


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E(A, A)E(B, A)P(ck, A)p(A) dA,


where E(A, A) ~, a,P(a,, A) and analogously for E( B, A). Now —2~E(A, A)[E(B,A)+E(B’, A)]

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Lemma. n-system state. For any two of Let the ~Pbean n systems, there entangled exists a projection, onto a direct product of states of the other n —2 systems, state. that leaves the two systems in an entangled


The proof proceeds by assuming the opposite, and finally contradicting the assumption that the n-sys-

since the sum is linear in each expectation value, and the expectation values take their extremal values at ±1. We multiply each term by p(A)P(ck, A)/P(ck) and integrate, obtaining a generalization ofthe CHSH inequality to measurements made on two systems, conditional on the results of measurements on a third,

tem state was entangled. We assume that any projection onto a direct product of states of the other n—2 systems leaves the remaining two systems in a product state. In particular, let e’>j be a given basis in the subspace of the jth system, and consider projections onto direct products of these basis vectors. We write

+E(A’, A) [E(B, A)

E(B’, A)] ~ 2,

—2~E(A,B, Ck) +E(A, B’, Ck) +E(A’,B,ck)—E(A’,B’,ck)~<2.

= !0>~ 10’>2,

On the other hand, the quantum correlation is obtamed by projecting the state ~Ponto the subspace of the third system having eigenvalue Ck, normalizing, and calculating the expectation value of the products AB, etc., with respect to this projected state. If the projected state is entangled then we know, from the results already obtained for two systems, that the quantum expectation value will violate the inequality just obtained. It remains to check that the projection of the three-system state W onto some particular state of the third system (coi~respondingto the eigenvalue ckof C) leaves the remainingtwo ~Y~tems in an entangled state. For an entangled state of three systems such a projection is always available, It is straightforward to extend the CHSH inequality to cover ensembles of n systems; we calculate correlations between two systems, conditional on a particular outcome for the other n —2 systems. The corresponding quantum predictions for an entangled state of n systems are obtained by fixing the eigenvalues of n —2 of the systems. The eigenvalues are fixed by an appropriate projection which must project onto a direct product of local subspaces. If the resulting two-system state is still entangled, then the quantum correlations for that state will violate the generalized CHSH inequality for n systems. Thus, we only need to prove that we can find a projection onto a direct product of local subspaces such that the resulting two-system state is still entangled. We present this fact as a lemma. 296


(16) where I0> and I0’>2 are the states of the two memaining systems. They are, by assumption, in a product state for any choice of indices i3, i4, 1,,, although the product state will in general depend on the indices. We show this dependence explicitly by writing ...,

I 0>1 = I 0 ( i3~i4~ i~) >1, I 0’> 2 = I 0’( i3, i4, i~) >2. ...~



Now suppose that in the direct product of basis vectons, we choose a different basis vector I e”3 >3 from the subspace 3. It is clear that either I0>~or 10’>2 must remain unchanged (up to a phase). For suppose that they both change. Then1 or I 0’> 2 is in fact independent of i 3. Repeating this argument for the other subspaces, we conclude that, despite the notation I 0 ( i3, 14, i~)>1 and I 0’ ( i3, i4, ~n) >2, each index actually appears in either I 0>1 on I 0’> 2 but not in both. By relabeling the subspaces, we can write ...,



I0(i3, i4,



1k±2, ..., ~n)>

I0’>2= I0’(ik±l, 2. (19) Now the identity operator can be resolved into a complete sum over orthogonal projections, so

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I ~P> =

I e”>



I e’~>4 I e’3>3

systems are in an entangled state, classical assumpof the separability fail, as Bohr in Podolsky his reply [10] to paradox presented by argued Einstein, and Rosen [11]. The superposition principle rules

3 I ~tions




IO’(~k+I, ..‘, ifl)>2Ie~>k+l...Ie



Ie’3> 3...Ie”’>k




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out classical separability! Elsewhere [12] we have discussed the implications of this generic quantum nonlocality.


(20) and we see that the state I ~ factors into a product, contradicting the assumption that it was an entangled state. The lemma is thus proved, and with it the extension of theorem 2 to entangled states of n systems. Theorem 3. For any entangled state of n macroscopically separated systems, a set of measurements

may be prescribed, for which the predictions of quantum mechanics are inconsistent with local realism. Between any pair of the systems, nonlocal correlations arise. Theorem 3 caps the progressive generalization of Bell’s inequality [2—5]. We have shown that nonlocal correlations, contradicting local realism, are not the exception but rather the rule for states that do not factor into local states. Not only are quantum theory and local realism incompatible, the nonlocality of quantum mechanics is generic. Whenever

We thank A. Elitzur, I. Unna and C. Uribe for comments. D.R. thanks the Niels Bohr Institute for hospitality while this work was in preparation.

References [1] J.S. Bell, Physics 1(1964)195. [2] V. Capasso, D. Fortunato and F. Selleri, mt. J. Mod. Phys. 7(1973)319. [3] N. Gisin, Phys. Lett. A 154 (1991) 201. [4] N. Gisin and A. Peres, Phys. Lett. A 162 (1992) 15. [5] R.F. Werner, unpublished result. [6] J.F. Clauser, M.A. Home, A. Shimony and R.A. Holt, Phys. Lett. 23(1969) 880. [7] Rev. S. Popescu and D. Rohrlich, Tel Aviv University preprint TAUP-1873-91. [8] D.M. Greenberger, MA. Home, A. Shimony and A. Zeilinger, Am. J. Phys. 58 (1990) 1131. [9] Y. Aharonov and D.Z. Albert, Phys. Rev. D 24 (1981) 359. [10] Bohr, Phys. Rev. 48 (1935) 696. Phys. Rev. 47 (1935) [11] N. A. Einstein, B. Podolsky and N. Rosen, 777. [12] A. Elitzur, S. Popescu and D. Rohrlich, Phys. Lett. A 162 (1992) 25; Tel Aviv University preprint TAUP-1848-90.