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.
Highlights
The Hypercube Picture 3.2.3 Explicit Analysis of Simple Inequalities 3.3 Explicit Constructions of the Hypergraph Cone 3.3.1 The 4-Party Hypergraph Cone 3.3.2 The 5-Party Hypergraph Cone
5 Discussion and Future Directions 5.1 The Hypergraph Cone and Quantum States 5.2 Searching for New Hypergraph Inequalities 5.3 Non-Holographic Bit Threads 5.4 Connections to Higher Derivative Gravity
Stacking the k bit strings of length m as the rows of a k × m matrix, 6Our usage of the word “state" suggests that hypergraphs can represent physical quantum states – we take this as an assumption for and provide some motivation for it in later sections. Evidence supporting that this prescription is still meaningful quantum mechanically will be 2-fold: for small party numbers, 1) we show that the resulting entropy vectors lie strictly inside the quantum entropy cone and 2) we are able to propose a plausible prescription to construct a quantum state given a hypergraph such that their entropies match exactly
Summary
The study of entanglement entropy in holography, as originated by the Ryu-Takayanagi (RT) formula [1] and its covariant generalization [2], has had a profound impact on the conceptualization and formulation of the holographic duality, from energy conditions [3,4,5,6] and ctheorems [7,8,9] to the emergence of spacetime itself [10,11,12,13]. There has been much work in attempting to characterize holographic entanglement and streamline the discovery of entropy inequalities by endowing entropy space with additional structures and studying useful reparameterizations thereof [24,25,26,27,28] Progress along these lines holds the potential to improve our understanding of the meaning of holographic constraints and aid in the computational tractability of finding and proving new holographic entanglement entropy inequalities.
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