Joint Probabilities Reproducing Three EPR Experiments On Two Qubits

Probabilities and correlations in EPR experiments. The EPR-Bohm-Aharonov [1] system of two correlated spin-half particles or qubits observed by spatially separated observers has often been used as an arena in which to probe fundamental questions about quantum theory. Typically there are four different configurations corresponding to four different experiments and one obtains no-go theorems on (i) the validity of Einstein’s local reality principle [2] and (ii) the existence of joint probabilities for noncommuting observables [3],[4],[5]. We demonstrate here a positive result concerning problem (ii), in the case of only three EPR experiments, by obtaining explicitly the complete set of joint probabilities of the relevant commuting and noncommuting observables. Complementarity — or the nonexistence of a joint probability for noncommuting observables — thus becomes a precise quantitative issue: it does not hold for 3 EPR experiments, it does hold for 4 EPR experiments. It should be stressed that Fine’s earlier construction of a particular joint probability for 4 EPR experiments only holds for those quantum states which do not violate Bell-CHSH inequalities. We obtain the most general joint probability (i) for three EPR experiments for arbitrary quantum states, as well as (ii) for four EPR experiments for those quantum states which obey Bell-CHSH inequalities.