Absolutely Maximal Contextual Correlations
Nripendra Majumdar, S. Aravinda
TL;DR
The paper defines absolutely maximal contextual correlations (AMCC) as correlations that are simultaneously maximally contextual ($CF=1$) and have maximal marginals, within the sheaf-theoretic framework of contextuality. It provides concrete realizations: in the bipartite $(2,2,2)$ case, PR boxes are AMCCs, while in the tripartite $(3,2,2)$ case GHZ-based correlations also yield AMCCs with uniformly random bipartite marginals, and it demonstrates the existence of infinite families via parity-check and CSP constructions. It develops two complementary construction methods—the CSP-model and parity-check approach—showing that most parity-constructed instances are AMCCs and that CSP can generate additional strongly contextual but non-AMCC correlations; it also presents a reduced eight-parameter and a broader 26-parameter formalism for these models. The work demonstrates practical relevance by linking AMCCs to device-independent secret sharing and randomness extraction, where maximal marginals enable optimal min-entropy across subsystems, and discusses implications for the interplay between nonlocality, contextuality, and randomness resources in multipartite quantum and pre-quantum theories.
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.