On the foundations and extremal structure of the holographic entropy cone

Abstract

The holographic entropy cone (HEC) is a polyhedral cone first introduced in the study of a class of quantum entropy inequalities. It admits a graph-theoretic description in terms of minimum cuts in weighted graphs, a characterization which naturally generalizes the cut function for complete graphs. Unfortunately, no complete facet or extreme-ray representation of the HEC is known. In this work, starting from a purely graph-theoretic perspective, we develop a theoretical and computational foundation for the HEC. The paper is self-contained, giving new proofs of known results and proving several new results as well. These are also used to develop two systematic approaches for finding the facets and extreme rays of the HEC, which we illustrate by recomputing the HEC on $5$ terminals and improving its graph description. We also report on some partial results for $6$ terminals. Some interesting open problems are stated throughout.

34 pages, 2 figures, 3 tables. Revised to add partial results for 6 terminals and to clarify various points

References in corpus (7)

• Quantum Information in Holographic Duality
• The holographic entropy arrangement
• A holographic inequality for $N=7$ regions
• The Symmetrized Holographic Entropy Cone
• Holographic Cone of Average Entropies
• The holographic entropy cone from marginal independence
• The quantum marginal independence problem

Cited by in corpus (16)

• Quantum Extremal Surfaces and the Holographic Entropy Cone
• Holographic Entropy Inequalities and Multipartite Entanglement
• The Symmetrized Holographic Entropy Cone
• Topological Link Models of Multipartite Entanglement
• Testing holographic entropy inequalities in 2+1 dimensions
• Properties of the contraction map for holographic entanglement entropy inequalities
• More on the upper bound of holographic n-partite information
• Towards a complete classification of holographic entropy inequalities
• Entanglement Wedge Cross Section Inequalities from Replicated Geometries
• A gap between holographic and quantum mechanical extreme rays of the subadditivity cone
• Improved proof-by-contraction method and relative homologous entropy inequalities
• Beyond the Holographic Entropy Cone via Cycle Flows
• Purely GHZ-like entanglement is forbidden in holography
• On the completeness of contraction map proof method for holographic entropy inequalities
• Upper bound of holographic entanglement entropy combinations
• Exploring the holographic entropy cone via reinforcement learning