Correlations and entanglement in probability theory

Abstract

We generalize the classical probability frame by adopting a wider family of random variables that includes nondeterministic ones. The frame that emerges is known to host a ''classical'' extension of quantum mechanics. We discuss the notion of probabilistic correlation and show that it includes two kinds of correlation: a classical one, which occurs for both deterministic and indeterministic observables, and a nonclassical one, which occurs only for indeterministic observables. The latter will be called probabilistic entanglement and represents a property of intrinsically random systems, not necessarily quantum. It appears possible to separate the two kinds of correlation and characterize them by numerical functions which satisfy a simple product rule.