Necessary and Sufficient Conditions for an Extended Noncontextuality in a Broad Class of Quantum Mechanical Systems

The notion of (non)contextuality pertains to sets of properties measured one subset (context) at a time. We extend this notion to include so-called inconsistently connected systems, in which the measurements of a given property in different contexts may have different distributions, due to contextual biases in experimental design or physical interactions (signaling): a system of measurements has a maximally noncontextual description if they can be imposed a joint distribution on in which the measurements of any one property in different contexts are equal to each other with the maximal probability allowed by their different distributions. We derive necessary and sufficient conditions for the existence of such a description in a broad class of systems including Klyachko-Can-Binicioğlu-Shumvosky-type (KCBS), EPR-Bell-type, and Leggett-Garg-type systems. Because these conditions allow for inconsistent connectedness, they are applicable to real experiments. We illustrate this by analyzing an experiment by Lapkiewicz and colleagues aimed at testing contextuality in a KCBS-type system.

Figure 1

A schematic representation of a cyclic (single-cycle) system of rank $n>1$. The properties $q_1,\dots ,q_n,q_1$ form a circle, any two successive properties $(q_i,q_{i\oplus 1})$ form a context, denoted $c_i$ ($\oplus$ is clockwise shift $1\mapsto 2\mapsto \cdots \mapsto n\mapsto 1$). In a given context $c_i$ the random variable representing $q_i$ is denoted $R_i^i$, and the one representing $q_{i\oplus 1}$ is denoted $R_{i\oplus 1}^i$. Each property $q_i$, therefore, is represented by two random variables: $R_i^i$ (when $q_i$ is measured in context $c_i$) and $R_i^{i\ominus 1}$ (when $q_i$ is measured in context $c_{i\ominus 1}$). The pair $(R_i^{i\ominus 1},R_i^i)$ is the connection for $q_i$, and the pair $(R_i^i,R_{i\oplus 1}^i)$ represents the context $c_i$.