Probabilistic Contextuality in EPR/Bohm-type Systems with Signaling Allowed

We propose a principled way of defining and measuring contextuality in systems with deterministic inputs and random outputs. We illustrate it on systems with two binary inputs and two binary random outputs, the prominent example being the system of two entangled spin-half particles with each particle’s spins (random outputs) being measured along one of two directions (inputs). It is traditional to say that such a system exhibits contextuality when it violates Bell-type inequalities. Derivations of Bell-type inequalities, however, are based on the assumption of no-signaling, more generally referred to as marginal selectivity: the distributions of outputs (spins) in Alice’s particle do not depend on the inputs (directions chosen) for Bob’s particle. In many applications this assumption is not satisfied, so that instead of contextuality one has to speak of direct cross-influences, e.g., of Bob’s settings on Alice’s spin distributions. While in quantum physics direct cross-influences can sometimes be prevented (e.g., by space-like separation of the two particles), in other applications, especially in behavioral and social systems, marginal selectivity almost never holds. It is unsatisfying that the highly meaningful notion of contextuality is made inapplicable by even slightest violations of marginal selectivity. Our new approach rectifies this: it allows one to define and measure contextuality on top of direct cross-influences, irrespective of whether marginal selectivity (no-signaling condition) holds. For systems with two binary inputs and two binary random outputs, contextuality means violation of the classical CHSH inequalities in which the upper bound 2 is replaced with 2(1+∆0), where ∆0 is a measure of deviation from marginal selectivity.