Monogamy of Mutual Information in Graph States

TL;DR

This paper investigates the conditions under which the monogamy of mutual information (MMI) fails in graph states, linking these failures to specific graph substructures and providing a conjecture and proof for certain cases.

Contribution

It introduces a forbidden-subgraph conjecture for MMI violations in graph states and proves it for a family of star-like graphs, extending understanding of quantum entropy inequalities.

Findings

Identifies a connection between MMI violations and four-star subgraphs in graph states.

Constructs star-like graphs that violate specific MMI instances.

Shows that MMI failure cannot be fully explained by the studied graph structures.

Abstract

The monogamy of mutual information (MMI) is a quantum entropy inequality that enforces the non-positivity of tripartite information. We investigate the failure of MMI in graph states as a forbidden-subgraph phenomenon, conjecturing that every MMI-violating graph state is local-Clifford equivalent to one whose graph contains a four-star subgraph. We construct a family of star-like graphs whose states fail a specific class of MMI instances, and extend this analysis to general star topologies. Deriving adjacency matrix constraints that fix the MMI evaluation for these instances and interpreting them physically, we prove the forbidden-subgraph conjecture for this family of graphs. Finally, through an exhaustive search over graph representatives for all $8$-qubit stabilizer entropy vectors, we establish that MMI failure is not reducible to the cases within our scope.