Quantum Contextuality

TL;DR

This paper investigates the structure and properties of quantum contextual sets using hypergraph-based inequalities, employing automated algorithms to generate and analyze these sets across various dimensions for applications in quantum information.

Contribution

It introduces a systematic method for generating and analyzing quantum contextual hypergraphs and evaluates the validity of different inequalities for discrimination.

Findings

Certain inequalities are inappropriate for noncontextuality testing.

Some hypergraph-based inequalities are not valid noncontextuality inequalities.

Automated algorithms successfully generate contextual hypergraphs in multiple dimensions.

Abstract

Quantum contextual sets have been recognized as resources for universal quantum computation, quantum steering and quantum communication. Therefore, we focus on engineering the sets that support those resources and on determining their structures and properties. Such engineering and subsequent implementation rely on discrimination between statistics of measurement data of quantum states and those of their classical counterparts. The discriminators considered are inequalities defined for hypergraphs whose structure and generation are determined by their basic properties. The generation is inherently random but with the predetermined quantum probabilities of obtainable data. Two kinds of statistics of the data are defined for the hypergraphs and six kinds of inequalities. One kind of statistics, often applied in the literature, turn out to be inappropriate and two kinds of inequalities turn out not to be noncontextuality inequalities. Results are obtained by making use of universal automated algorithms which generate hypergraphs with both odd and even numbers of hyperedges in any odd and even dimensional space - in this paper, from the smallest contextual set with just three hyperedges and three vertices to arbitrarily many contextual sets in up to 8-dimensional spaces. Higher dimensions are computationally demanding although feasible.