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 , 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  and Gisin  showed that any entangled state of two spin-I particles violates a Bell inequality. In a recent paper of Gisin and Peres  this resuit extends to pairs of systems with effective Hilbert Supported by the US—Israel Binational Science Foundation.
spaces of arbitrary dimension . 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
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
29 June 1992
<°~ ~ = 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 ‘