Space-Time in the Quantum World

Abstract

Consider pairs of electrons produced in the singlet state and allowed to separate to a very great distance. We know from the work of Bell that no theory can predict violations of Bell’s inequality for spin measurements on those pairs if it satisfied two conditions. The first (condition a) is that the theory ascribe either a single initial state or a convex sum of states to the ensemble of pairs such that the initial states are statistically independent of the spin measurements later carried out on the electrons. The second (condition b) is most easily understood if expressed differently for deterministic and for stochastic theories. For a deterministic theory we require (condition b′) that the theory determine the results of each measurement solely on the basis of the initial state and the details of the measurement carried out on that particle. For a stochastic theory we demand (condition b*) that the theory assign probabilities for measurement results based solely on the initial state and the measurement carried out on a single electron, which probabilities are unchanged when one conditionalizes on the measurements and results obtained on the other particle in the pair. (Conditions b′ and b* are really the same condition, expressed in the one case appropriately to a deterministic theory, in the other for an indeterministic one. A theory which violates b′ also violates b* since conditionalizing on information about the measurement carried out on the second electron can render the result of the first measurement certain.) Einstein et al. (1935) had already pointed out the impossibility of a non-deterministic theory which obeys a and b* to recover the predictions of quantum theory, and so argued in favor of a deterministic theory. Bell showed that no deterministic theory obeying a and b′ could recover all of the predictions of quantum mechanics. The later work of Greenberger et al. (1989) showed that the reference to ensembles is otiose: there can be individual triples of particles such that no initial state of type a can recover all of the predictions of quantum mechanics if the theory is of type b′ or b*.