Correlation hypergraph: a new representation of a quantum marginal independence pattern

TL;DR

This paper introduces correlation hypergraphs as a novel, efficient way to represent quantum marginal independence patterns, linking quantum information theory with holography and graph theory to derive new structural insights.

Contribution

The paper presents a new hypergraph-based representation of quantum marginal independence patterns, improving the understanding of their structure and compatibility with strong subadditivity.

Findings

Correlation hypergraphs generalize entanglement wedge relations in holography.

Derived a necessary, efficiently testable condition for entropy vector realizability in holography.

Provided insights into the combinatorial structure of SSA-compatible quantum systems.

Abstract

We continue the study of the quantum marginal independence problem, namely the question of which faces of the subadditivity cone are achievable by quantum states. We introduce a new representation of the patterns of marginal independence (PMIs, corresponding to faces of the subadditivity cone) based on certain correlation hypergraphs, and demonstrate that this representation provides a more efficient description of a PMI, and consequently of the set of PMIs which are compatible with strong subadditivity. We then show that these correlation hypergraphs generalize to arbitrary quantum systems the well known relation between positivity of mutual information and connectivity of entanglement wedges in holography, and further use this representation to derive new results about the combinatorial structure of collections of simultaneously decorrelated subsystems specifying SSA-compatible PMIs. In the context of holography, we apply these techniques to derive a necessary condition for the realizability of entropy vectors by simple tree graph models, which were conjectured in arXiv:2204.00075 to provide the building blocks of the holographic entropy cone. Since this necessary condition is formulated in terms of chordality of a certain graph, it can be tested efficiently.