Polyteam logics for quantum contextuality

Abstract

We introduce logics of dependence and independence over polyteams (that is, databases with multiple relation schemes), in a similar way as in arXiv:1704.02158v3 [cs.LO]. On this basis, we develop a logical framework in which we can express (in formulas) relevant properties about the statistics of a repeated quantum experiment. In this framework, we formalize Bell's theorem, and the Kochen-Specker theorem as failure of logical entailments, as it was suggested in arXiv:2107.10817v3 [math.LO]. The key feature of polyteam logics in action here is their power to express both local and global properties, with respect to the set of possible measurements. The point of this is at least twofold. Firstly, as in arXiv:1102.0264v7 [quant-ph], it gives a high-level (information-theoretic) understanding of these physical phenomena, that is not intrinsic to the physical objects involved. Secondly, since logical entailment is often (partially) axiomatizable, this framework incites us to tackle the question of contextuality computationally. We also define semiring-weighted polyteams along with logics on them, and use them to recover the probabilistic versions of the contextuality theorems mentioned above.