The Holographic Entropy Arrangement

• Published in: Fortschritte der Physik 2019-04, Vol.67 (4), p.n/a
• Main Authors: Hubeny, Veronika E., Rangamani, Mukund, Rota, Massimiliano
• Format: Article
• Language: English
• Subjects: Algebra; Combinatorial analysis; Entanglement; Entropy; Holography; Hyperplanes; multipartite entanglement; Polyhedra; Subsystems; Time dependence
• ISSN: 0015-8208; 1521-3978
• DOI: 10.1002/prop.201900011

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. Here a convenient framework is developed 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. It will be proven that, for an arbitrary number of parties, the procedure proposed identifies a finite set of entropic information quantities that is conveniently represented geometrically in the form of an arrangement of hyperplanes. This leads to define the holographic entropy arrangement, whose algebraic and combinatorial aspects we explore in detail. Using the framework, one derives 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. It will be illustrated how to obtain the polyhedron by winnowing do