Absolutely Maximal Contextual Correlations

TL;DR

This paper introduces a new framework for understanding maximal contextual correlations in quantum systems, defining absolutely maximal correlations with applications to secret sharing and randomness extraction.

Contribution

It formalizes the concept of absolutely maximal contextual correlations using sheaf-theoretic methods and constructs various examples, including extensions of PR-boxes and non-AMCC correlations.

Findings

Defined the sheaf-theoretic metric for contextuality (CF).

Constructed infinite families of AMCC correlations.

Demonstrated applications in secret sharing and randomness extraction.

Abstract

The foundational work by Bell led to an interest in understanding non-local correlations that arise from entangled states shared between distinct, spacelike-separated parties, which formed a foundation for the theory of quantum information processing. We investigate the question of maximal correlations analogous to the maximally entangled states defined in the entanglement theory of multipartite systems. To formalize this, we employ the sheaf-theoretic framework for contextuality, which generalizes non-locality. This provides a new metric for correlations called contextual fraction (CF), which ranges from 0 (non-contextual) to 1 (maximally contextual). Using this, we have defined the absolutely maximal contextual correlations (AMCC), which are maximally contextual and have maximal marginals, which captures the notion of absolutely maximal entangled (AME) states. The Popescu-Rohrlich (PR) box serves as the bipartite example, and we construct various extensions of such correlations in the tripartite case. An infinite family of various forms of AMCC is constructed using the parity check method and the constraint satisfiability problem (CSP) scheme. We also demonstrate the existence of maximally contextual correlations, which do not exhibit maximal marginals, and refer to them as non-AMCC. The results are further applied to secret sharing and randomness extraction using AMCC correlations.