Four Party Absolutely Maximal Contextual Correlations

TL;DR

This paper introduces a new class of four-party correlations called absolutely maximal contextual correlations (AMCC), extending the concept of maximality in multipartite systems using sheaf theory and CSP methods.

Contribution

It defines AMCC in multipartite systems, extends the framework to four parties, and constructs explicit examples using CSP and parity check techniques.

Findings

AMCC generalizes maximal contextuality to four-party systems.

No AME state exists for four qubits, highlighting differences from AMCC.

Explicit non-AMCC correlations are constructed within the CSP framework.

Abstract

The Kochen Specker theorem revealed contextuality as a fundamental nonclassical feature of nature. Nonlocality arises as a special case of contextuality, where entangled states shared by space like separated parties exhibit nonlocal correlations. The notion of maximality in correlations, analogous to maximal entanglement, is less explored in multipartite systems. In our work, we have defined maximal correlations in terms of contextual models, which are analogous to absolutely maximally entangled (AME) states. Employing the sheaf theoretic framework, we introduce maximal contextual correlations associated with the corresponding maximal contextual model. The formalism introduces the contextual fraction CF as a measure of contextuality, taking values from 0 (noncontextual) to 1 (fully contextual). This enables the formulation of a new class of correlations termed absolutely maximal contextual correlations (AMCC), which are both maximally contextual and maximal marginals. In the bipartite setting, the canonical example is the Popescu Rohrlich (PR) box, while in the tripartite case, it includes Greenberger Horne Zeilinger (GHZ) correlations and three way nonlocal correlations. In this work, we extend these findings to four party correlations. Notably, no AME state exists for four qubits, which introduces a subtle difference between AMCC and AME. The construction follows the constraint satisfaction problem (CSP) and parity check methods. In particular, the explicit realization of a non AMCC correlation that is maximally contextual yet not maximal marginal is obtained within the CSP framework.