INTRODUCTION

The main purpose of this chapter is to prove a lemma about random variables, and then to apply this lemma to the characterization of local theories of hidden variables by Bell (1964, 1966) and Wigner (1970), which are focused around Bell's inequality. We use the results of the lemma in two different ways. The first is to show that the assumptions of Bell and Wigner can be weakened to conditional statistical independence rather than conditional determinism because determinism follows from conditional independence and the other assumptions that are made about systems of two spin-½ particles.

The second direction is to question the attempt of Clauser and Home (1974) to derive a Bell-type inequality for local stochastic theories of hidden variables which use an assumption of conditional statistical independence for observables.

The main thrust of our analysis obviously arises from the probabilistic lemma we prove. Roughly speaking, this lemma asserts that if two random variables have strict correlation, that is, the absolute value of the correlation is one, and it is in addition assumed that their expectations are conditionally independent when a third random variable λ is given, then the conditional variance of X and Y given λ is zero. In other words, given the hidden variable λ the observables X and Y are strictly determined. The lemma itself, of course, depends on no assumptions about quantum mechanics. It may be regarded as a limitation on any theories that assume both strict correlation between observables and their conditional independence on the basis of some prior or hidden variable.