Generalizing Bell-type and Leggett-Garg-type Inequalities to Systems with Signaling

Contextuality means non-existence of a joint distribution for random variables recorded under mutually incompatible conditions, subject to certain constraints imposed on how the identity of these variables may change across these conditions. In simple quantum systems contextuality is indicated by violations of Bell-type or Leggett-Garg-type inequalities. These inequalities, however, are predicated on the assumption of no-signaling, defined as invariance of the distributions of measurement results with respect to other (e.g., earlier in time) measurements' settings. Signaling makes the inequalities inapplicable: a non-signaling system with any degree of contextuality, however high, loses any relation to this concept as soon as it exhibits any degree of signaling, however small. This is unsatisfactory. We describe a principled way of defining and measuring contextuality in arbitrary systems with random outputs, whether signaling is absent or present.