Dichotomic Functions and Bell’s Theorems

Warren K. Leffler

Marek Żukowski, Časlav Brukner

Abstract We present a single general Bell inequality that summarizes all possible Bell type constrains on the correlation function for the N‐particle system (i.e., it is a necessary and sufficient condition for the existence of local realistic models for the correlation function). It is applicable in the case when measurements on each particle can be chosen between two arbitrary dichotomic observables. We also obtain a necessary and sufficient condition for an arbitrary N‐qubit mixed state to violate this general Bell inequality.

А. В. Белинский

Marek Żukowski, Časlav Brukner

We derive a single general Bell inequality which is a sufficient and necessary condition for the correlation function for N particles to be describable in a local and realistic picture, for the case in which measurements on each particle can be chosen between two arbitrary dichotomic observables. We also derive a necessary and sufficient condition for an arbitrary N-qubit mixed state to violate this inequality. This condition is a generalization and reformulation of the Horodecki family condition for two qubits.

Andrea Aiello

Bell's theorem supposedly demonstrates an irreconcilable conflict between quantum mechanics and local, realistic hidden variable theories. In this paper we show that all experiments that aim to prove Bell's theorem do not actually achieve this goal. Our conclusions are based on a straightforward statistical analysis of the outcomes of these experiments. The key tool in our study is probability theory and, in particular, the concept of sample space for the dichotomic random variables that quantifies the outcomes of such experiments. We also show that an experimental proof of Bell's theorem is not, in principle, impossible, but it would require a completely different experimental apparatus than those commonly used to allegedly achieve this objective. The main consequence of our work is that we cannot dismiss local realistic hidden variable theories on the basis of currently available experimental data.