The holographic entropy arrangement

TL;DR

This paper introduces a geometric framework called the holographic entropy arrangement to analyze multipartite entanglement, deriving new information quantities and entropy inequalities that match holographic entropy cones.

Contribution

It develops a novel geometric and algebraic framework for characterizing multipartite entanglement using hyperplane arrangements, leading to new entropy quantities and inequalities.

Findings

Derived three new information quantities for four parties.

Established a new infinite family of quantities for any number of parties.

Confirmed the holographic entropy cone for 4 and 5 parties.

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 arXiv:1808.07871, 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.