Bell's theorem : experimental tests and implications - Physics at ...

1888 F Clausw and A Shimonythe result A (EPR’s premise (ii)). Since the quantum-mechanical state Y does notdetermine the result of an individual measurement, this fact (via EPRs argument)suggests that there exists a more complete specification of the state in which thisdeterminism is manifest. We denote this state by the single symbol A, although itmay well have many dimensions, discrete and/or continuous parts, and different partsof it interacting with either apparatus, etc. Presumably the quantum state Y is arelated partial specification of this state. We thus define a deterministic hiddenvariablestheory as any physical theory which postulates the existence of states of asystem, for which the observables of quantum mechanics always have definite values.Let A be the space of the states X for an ensemble comprised of a very large numberof the observed systems. We make no restrictions as to what type of space this is,nor to its dimensionality, nor do we require linearity for operations with it, but ofcourse we require that a set of Bore1 subsets of A be defined, so that probabilitymeasures can be defined upon it. We represent the distribution function for the statesA on the space A by the symbol p. For this ensemble we take p to have norm one:JA dp= 1. (3.3)In a deterministic hidden-variables theory the observable Ai. B6 has a definitevalue (Ad. Ba) (A) for the state A. For these theories Bell defined locality as follows:a deterministic hidden-variables theory is local if for all d and 6 and all A E A we have:(AL.Ba) (A)=A,(X).Bs(X). (3 *4)That is, once the state X is specified and the particles have separated, measurements ofA can depend only upon X and d but not 6. Likewise measurements of B depend onlyupon h and 6, Any reasonable physical theory that is realistic and deterministic andthat denies the existence of action-at-a-distance is local in this sense. (More generaldefinitions of ‘local’ will be considered in 93.3.) For such theories the expectationvalue of Ai.Ba is then given byE(a, b)= JA Aa(A)Bs(X) dp. (3.5)Bell’s (1965) proof of the theorem consists of showing that if the locality condition (3.4)and the condition (3.2) for partial agreement with quantum mechanics are bothsatisfied, then the expectation values satisfy a simple inequality. This inequality isthen an alternative prediction to that by quantum mechanics for the expectation valueof A,.Bs. The predictions made by this inequality are quantitatively different fromthose of equation (3.1).The demonstration is straightforward. Equation (3.2) can hold if and only ifAd( A) = - Bi( A) (3.6)holds for all h E A. Using equation (3.6) we calculate the following function, whichinvolves three different possible orientations of the analysers :E(d, &)-E(d, e)= - JA [A;(h)As(h)-A,(X)A,(X)] dp= - JA Aa(h)As(h)[l -&(h)&(A)]dp.Since A, B = F 1, this last expression can be written :