Volume 118, number 1
PHYSICS LETTERS A
22 September 1986
BELL'S INEQUALITIES AND THEIR RELEVANCE T O T H E P R O B L E M O F N O N L O C A L I T Y IN Q U A N T U M M E C H A N I C S L.C.B. R Y F F
Unwersidade Federal do Rio de Janeiro, Instituto de ~sica, Cidade Unwersithria, llha do FundSo, Rio de Janeiro 21941, Brazil Received 22 May 1986; accepted for publication 29 July 1986
A straightforward derivation of the coefficient o)~'~orrelation of polarization present in Bell's inequalities is given, where appeal to an action-at-a-distance interaction is explicitly made. This demonstrates that nonlocal hidden variables theories may be compatible with quantum mechanics, and the relevance of Bell's inequalities to the problem of nonlocality in such theories. It follows that, despite some recent criticism, the locality assumption plays an essential role in the derivation of Bell's inequalities. Bell's inequalities can be written  as
I E ( a , b) - E ( a , b') + E ( a ' , b) + E ( a ' , b ' ) l ~< 2,
where E ( a , b) is the coefficient of correlation of polarization with polarizers in orientation a and b. Usually, to calculate E ( a , b) one must know how to calculate the expectation value of the p r o d u c t of two operators for a two-photon spin function . This contrasts with the simplicity and general features of the deduction of (1). It seems desirable, from a conceptual as well as from a pedagogical point of view, to show that we can calculate E ( a , b) from very basic and elementary facts, without appealing to any elaborate theory. As we will see, this can easily be accomplished by taking into account the collapse of the state vector. This allows us to explicitly show that Bell's inequalities can be violated by nonlocal hidden variables theories that assume an action-at-a-distance interaction. Although this can be considered an almost trivial result, it is important to stress it since there still seem to be some misconceptions on this matter [2,3]. We hope our result will make it definitively clear that Bell's inequalities cannot be derived for nonlocal hidden variables theories. This indeed leaves open the possibility that q u a n t u m mecha-
nics may be reproduced by a nonlocal hidden variables theory. F r o m the following, therefore, it should be clear that the mere existence o f hidden variables is not sufficient to yield Bell's inequalities. Hence, contrary to local ones, nonlocal hidden variables theories may be compatible with quantum mechanics. Local and nonlocal theories not being on an equal footing it follows that Bell's inequalities are extremely relevant to the problem of nonlocality in hidden variables theories, In a typical experiment performed to test (1) , two correlated photons, ~ and u2, are emitted in opposite directions. When a p h o t o n reaches a polarizer it is placed in a polarization state which can be parallel or perpendicular to the polarizer orientation. The collapse of the state vector is formally equivalent to assuming that this information is transmitted by an action-at-a-distance interaction to the other photon, which is then placed in the same polarization state. We will use this analogy to determine the q u a n t u m mechanical value of E ( a , b). It is assumed - obviously this can always be done - that the photons do not reach the polarizers exactly at the same time. For calculation purposes, in the following we will make the supposition that p h o t o n ~,~ reaches the polarizer in orientation a before u2 reaches that in orientation b. Since a polarizer transmits only one-half of the
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Volume 118, number 1
PHYSICS LETTERS A
intensity of an incident unpolarized light, P + ( " ) = P_Ca) = P+(b) = P_(b) = ~,
finding + 1 for r~
where P+(a), etc., are the probabilities of single measurements, i.e. P+(a) is the probability of finding + 1 (polarization parallel to the polarizer orientation) for u 1, etc. Using our action-at-a-distance analogy, p++(a, b) and p+ (a, b), respectively the probabilities of finding + 1 and - 1 for ~2 when +1 has previously been found for vl, are given by the Malus law. Naturally, p__(a, b) = p + + ( a , b) and p = + ( a , b)=p+_(a, b). Hence,
p++(a, b ) = p
( a , b ) = cosZ(a, b),
p+ (,,, b)=p_+Ca, t,)= sin2Ca, b).
22 September 1986
If P+ (a, b) is the probability of finding +1 for v+ and - 1 for u2, then (2) and (3) give us
1"+(,) = fAp(X , a ) p ( X ) dX = ½.
Of course, we may conceive as many hidden variables models as we wish that satisfy relation (6). According to our action-at-a-distance assumption, the probability p(X, a, b) of finding +1 for v~ and v2 when the photons emerge from the source in the state ~, is simply the product p ( X , ,,, b ) = p ( ~ , ,
where p(a, b) is the probability of finding + 1 for v2 when (as a consequence of the previous measurement on vl) it is linearly polarized in the direction a and the polarizer is in orientation b. According to the Malus law, p ( , , , b ) = cos2(,,, b).
Hence, using (7), (8) and (6), we obtain
P++(a, b)=P+(a)p++(a, b)
P++(,,, b)= f f c x , ,1, b)0CX) dX
= P--C,,, b) = P Ca)p_ (,,, b) = ½cos2(,,, b),
= ~cos:(a, o),
P+ C,,, b)= p+(a)p+ Ca, t,) = P +(~, b) = P ( o ) p _ + ( ~ , ~,) = ½sin2(a, b).
Using (4) we calculate the coefficient of correlation of polarization
E(a, b) = P++(a, b) + P
- p + _ ( a , b ) - P +C~, b) = cos 2 ( a , b),
in agreement with the standard quantum mechanical calculations . For the sake of completeness we will rederive the previous result explicitly introducing the notion of hidden variables. Let p(X) d~, be the probability of finding v~ and v2 in a hidden variable state comprised between ~, and X + dX . If p(X, a) is the probability of finding + 1 for v l when it comes from the source in the state ),, then, we must have for the total probability of
in agreement with (4). By means of a similar procedure the corresponding expressions for P+ (a, b), etc., as well as formula (5) can also be found. Therefore, the violation of Bell's inequalities can simply be interpreted within the context of a realistic philosophy if we assume an action-at-adistance interaction. We may conclude that in Bell's original derivation the locality assumption is not superfluous at all since quantum mechanics and nonlocal hidden variables theories are not necessarily incompatible.
References  J.F. Clauser and A. Shimony, Rep. Prog, Phys. 41 (1978) 1881.  W.M. de Muynck, Phys. Lett. A 114 (1986) 65.  A, Fine, Phys. Rev. Lett. 48 (1982) 291.  A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 49 (1982) 91.