Hypercyclic systems of measurements and patterns of contextuality

TL;DR

This paper introduces hypercyclic systems of measurements, generalizing cyclic systems, to study patterns of contextuality by analyzing four distinct measures that are not functionally related, thus capturing different aspects of contextuality.

Contribution

The paper proposes hypercyclic systems as a new framework that allows systematic study of contextuality patterns using multiple measures simultaneously.

Findings

Hypercyclic systems generalize cyclic systems while maintaining parametrization.

The four measures of contextuality are independent and not functions of each other within hypercyclic systems.

Hypercyclic systems enable comprehensive analysis of contextuality patterns.

Abstract

We consider four measures of contextuality, chosen for being based on the fundamental properties of the notion of contextuality, and for being applicable to arbitrary systems of measurements, both without and with disturbance. We have previously shown that no two of them are functions of each other: as systems of measurements change, either of them can change, while the other remains constant. This means that they measure different aspects of contextuality, and we proposed that rather than picking just one measure of contextuality in one specific sense, one could use all of them to characterize a contextual system by its pattern of contextuality. To study patterns of contextuality, however, one needs a systematic way of varying systems of measurements, which requires their convenient parametrization. We have convenient parametrization within the class of cyclic systems that have played a dominant role in the foundations of quantum mechanics. However, they cannot be used to study patterns of contextuality, because within this class the four measures of contextuality have been shown to be proportional to each other. In this concept paper, we introduce hypercyclic systems of measurements. They generalize cyclic systems while preserving convenient parametrization. We show that within this class of systems, the same as for systems at large, no two of the measures of contextuality are functions of each other. This means that hypercyclic systems can be used to study patterns of contextuality.