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.

Topological Link Models of Multipartite Entanglement

“…To motivate the construction of the link model, we share an explicit quantum state whose subsystem entropies can be represented by the link model yet provably fail to be realizable using hypergraph models [5,10]. The pertinent entropy vector, henceforth referred to as ray 15 following [10], comes from a particular 5-party mixed graph state, 6 where the vertices denote spin systems and edges represent Ising-type interactions [18]. Explicitly, ray 15 reads…”

“…Further explorations of their connection to the latter revealed that the hypergraph model was able to encode the entropies of a subset of stabilizer states [8,9], and that this containment was proper [10]. The latter result relied on the discovery of a particular stabilizer state whose entropy vector, referred to as ray 15 in [10], violates an inequality proven to be valid for all hypergraphs in the hypergraph model. By contrast, another avenue of study has been the study of topology as an efficient encoding of certain patterns of entanglement.…”

“…1 The hypergraph model also provided interesting new results intimately related to the properties of ranks of linear subspaces and the entropies of stabilizer states in quantum mechanics [5]. Further explorations of their connection to the latter revealed that the hypergraph model was able to encode the entropies of a subset of stabilizer states [8,9], and that this containment was proper [10]. The latter result relied on the discovery of a particular stabilizer state whose entropy vector, referred to as ray 15 in [10], violates an inequality proven to be valid for all hypergraphs in the hypergraph model.…”

Hypergraph States in SU(N)1, N odd prime, Chern-Simons Theory

“…We thank Zi-Wen Liu for clarifying the difference between hypergraph / stabilizer cones [28][29][30][31], and hypergraph / stabilizer states, and for emphasizing that universal quantum computation models for SU(2) 3 and SU(3) 2 have gates / actions which are magical.…”

Improved proof-by-contraction method and relative homologous entropy inequalities

“…Therefore, what is the general structure of HEC and how to prove HEC inequalities become a key to the whole story. Over the last few years many progresses have been made in this direction, see for example the incomplete list [8][9][10][11][12][13]. Among them a proof technique for holographic entropy inequalities called proof-by-contraction was developed in [7].…”