The quantum entropy cone of hypergraphs

Abstract

In this work, we generalize the graph-theoretic techniques used for the holographic entropy cone to study hypergraphs and their analogously-defined entropy cone. This allows us to develop a framework to efficiently compute entropies and prove inequalities satisfied by hypergraphs. In doing so, we discover a class of quantum entropy vectors which reach beyond those of holographic states and obey constraints intimately related to the ones obeyed by stabilizer states and linear ranks. We show that, at least up to 4 parties, the hypergraph cone is identical to the stabilizer entropy cone, thus demonstrating that the hypergraph framework is broadly applicable to the study of entanglement entropy. We conjecture that this equality continues to hold for higher party numbers and report on partial progress on this direction. To physically motivate this conjectured equivalence, we also propose a plausible method inspired by tensor networks to construct a quantum state from a given hypergraph such that their entropy vectors match.

Quantum Extremal Surfaces and the Holographic Entropy Cone

“…[28] for an upgraded version and ref. [58] for a generalization), but always formalized in the language of graph theory. Since graph models of holographic entanglement 8 have not played any role in our discussion so far, we will refrain from introducing them at this point.…”

Hyperthreads in holographic spacetimes

“…Our main goal will be to generalize bit threads in a way suitable for discussing multipartite entanglement. Taking inspiration from hypergraph states [5,8] we define a k-hyperthread or "k-thread" h as a connected codimension 1 subset of Σ which forms an embedded tree with between 1 and k − 1 branch points such that the cardinality #(h ∩ ∂Σ) = k (see figure 3). Given a partition A of ∂Σ we define the set of all hyperthreads whose endpoints connect to different boundary regions of A to be H A .…”

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Interlacing Results for Hypergraphs

“…Hypergraphs are a generalization of graphs in which edges can connect any number of vertices. They allow the modeling of bitcoin transactions [17], quantum entropies [2], chemical reaction networks [12], cellular networks [15], social networks [20], neural networks [6], opinion formation [16], epidemic networks [3], transportation networks [1]. Moreover, hypergraphs with real coefficients have been introduced in [13] as a generalization of classical hypergraphs where, in addition, each vertex-edge incidence is given a real coefficient.…”