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arXiv:2002.05317 · SciPost Physics

The quantum entropy cone of hypergraphs

Ning Bao, Newton Cheng, Sergio Hernández-Cuenca, Vincent Paul Su

Published Nov 10, 2020• 35 citations
arXivHTMLPDF

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.

DOI
10.21468/scipostphys.9.5.067

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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.
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.
The von Neumann entropy of pure quantum states and the min-cut function of weighted hypergraphs are both symmetric submodular functions. In this article, we explain this coincidence by proving that the min-cut function of any weighted hypergraph can be approximated (up to an overall rescaling) by the entropies of quantum states known as stabilizer states. We do so by constructing a novel ensemble of random quantum states, built from tensor networks, whose entanglement structure is determined by a given hypergraph. This implies that the min-cuts of hypergraphs are constrained by quantum entropy inequalities, and it follows that the recently defined hypergraph cones are contained in the quantum stabilizer entropy cones, which confirms a conjecture made in the recent literature.
Recent work has characterized the various inequalities that entanglement entropies represented by min-cuts on hypergraphs will satisfy. This collection, the hypergraph entropy cone, can be seen as a generalization of the holographic entropy cone which describes the entropies given by both min-cuts on 2-graphs and those of holographic states in AdS/CFT. In this article we describe a generalization of bit threads which allows us to describe max multiflows on hypergraphs. We further comment on its properties and interpretation in holography.
We develop a convenient framework for characterizing multipartite entanglement in composite systems, based on relations between entropies of various subsystems. This continues the program initiated in [1], of using holography to effectively recast the geometric problem into an algebraic one. We prove that, for an arbitrary number of parties, our procedure identifies a finite set of entropic information quantities that we conveniently represent geometrically in the form of an arrangement of hyperplanes. This leads us to define the holographic entropy arrangement, whose algebraic and combinatorial aspects we explore in detail. Using the framework, we derive three new information quantities for four parties, as well as a new infinite family for any number of parties. A natural construct from the arrangement is the holographic entropy polyhedron which captures holographic entropy inequalities describing the physically allowed region of entropy space. We illustrate how to obtain the polyhedron by winnowing down the arrangement through a sieve to pick out candidate sign-definite information quantities. Comparing the polyhedron with the holographic entropy cone, we find perfect agreement for 4 parties and corroborating evidence for the conjectured 5-party entropy cone. We work with explicit configurations in arbitrary (time-dependent) states leading to both simple derivations and an intuitive picture of the entanglement pattern.
We develop a reinforcement learning algorithm to study the holographic entropy cone. Given a target entropy vector, our algorithm searches for a graph realization whose min-cut entropies match the target vector. If the target vector does not admit such a graph realization, it must lie outside the cone, in which case the algorithm finds a graph whose corresponding entropy vector most nearly approximates the target and allows us to probe the location of the facets. For the N=3\sf N=3N=3 cone, we confirm that our algorithm successfully rediscovers monogamy of mutual information beginning with a target vector outside the holographic entropy cone. We then apply the algorithm to the N=6\sf N=6N=6 cone, analyzing the 6 "mystery" extreme rays of the subadditivity cone from arXiv:2412.15364 that satisfy all known holographic entropy inequalities yet lacked graph realizations. We found realizations for 3 of them, proving they are genuine extreme rays of the holographic entropy cone, while providing evidence that the remaining 3 are not realizable, implying unknown holographic inequalities exist for N=6\sf N=6N=6.
Abstract We develop a convenient framework for characterizing multipartite entanglement in composite systems, based on relations between entropies of various subsystems. This continues the program initiated in [1], of using holography to effectively recast the geometric problem into an algebraic one. We prove that, for an arbitrary number of parties, our procedure identifies a finite set of entropic information quantities that we conveniently represent geometrically in the form of an arrangement of hyperplanes. This leads us to define the holographic entropy arrangement, whose algebraic and combinatorial aspects we explore in detail. Using the framework, we derive three new information quantities for four parties, as well as a new infinite family for any number of parties. A natural construct from the arrangement is the holographic entropy polyhedron which captures holographic entropy inequalities describing the physically allowed region of entropy space. We illustrate how to obtain the polyhedron by winnowing down the arrangement through a sieve to pick out candidate sign‐definite information quantities. Comparing the polyhedron with the holographic entropy cone, we find perfect agreement for 4 parties and corroborating evidence for the conjectured 5‐party entropy cone. We work with explicit configurations in arbitrary (time‐dependent) states leading to both simple derivations and an intuitive picture of the entanglement pattern.