Quantum tests via inequalities for joint statistics

Abstract

In this work we derive statistical inequalities whose violation is equivalent to the impossibility of describing the data by a genuine joint probability distribution. So they are witnesses of the failure of a common probability space. We examine whether the most significant quantum quasidistributions violate these inequalities. We examine also whether these inequalitites are violated by the joint distributions derived form a noisy joint measurement. As a relevant example we find violations for the maximally mixed state, as well as cases where all system states violate them. This points to the idea that these results are more than a property of the system states, but are instead a property of the statistical structure of quantum mechanics.