A Gap Between the Hypergraph and Stabilizer Entropy Cones

Abstract

It was recently found that the stabilizer and hypergraph entropy cones coincide for four parties, leading to a conjecture of their equivalence at higher party numbers. In this note, we show this conjecture to be false by proving new inequalities obeyed by all hypergraph entropy vectors that exclude particular stabilizer states on six qubits. By further leveraging this connection, we improve the characterization of stabilizer entropies and show that all linear rank inequalities at five parties, except for classical monotonicity, form facets of the stabilizer cone. Additionally, by studying minimum cuts on hypergraphs, we prove some structural properties of hypergraph representations of entanglement and generalize the notion of entanglement wedge nesting in holography.

16 pages, 1 figure; more discussion of previous work, updated references, inclusion of appendix with explicit contraction map

References in corpus (2)

• Multi-party entanglement in graph states
• Entanglement entropy from SU(2) Chern-Simons theory and symmetric webs

Cited by in corpus (11)

• Quantum Extremal Surfaces and the Holographic Entropy Cone
• On the foundations and extremal structure of the holographic entropy cone
• The Symmetrized Holographic Entropy Cone
• Hypergraph min-cuts from quantum entropies
• Topological Link Models of Multipartite Entanglement
• Inequalities of Holographic Entanglement of Purification from Bit Threads
• Improved proof-by-contraction method and relative homologous entropy inequalities
• Bit threads on hypergraphs
• The Foliage Partition: An Easy-to-Compute LC-Invariant for Graph States
• Hypergraph States in SU(N)1, N odd prime, Chern-Simons Theory
• SU(N)1 Chern-Simons theory, the Clifford group, and Entropy Cone