The holographic entropy cone

Abstract

We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with smooth holographic dual geometries. For 2, 3, and 4 regions, we prove that the strong subadditivity and the monogamy of mutual information give the complete set of inequalities. This is in contrast to the situation for generic quantum systems, where a complete set of entropy inequalities is not known for 4 or more regions. We also find an infinite new family of inequalities applicable to 5 or more regions. The set of all holographic entropy inequalities bounds the phase space of Ryu-Takayanagi entropies, defining the holographic entropy cone. We characterize this entropy cone by reducing geometries to minimal graph models that encode the possible cutting and gluing relations of minimal surfaces. We find that, for a fixed number of regions, there are only finitely many independent entropy inequalities. To establish new holographic entropy inequalities, we introduce a combinatorial proof technique that may also be of independent interest in Riemannian geometry and graph theory.

The complexity of identifying Ryu-Takayanagi surfaces in AdS3/CFT2

“…Upon close examination, one can see that the proposition follows from the proof of lemma 3 in [5]. To show this, first recall the definition of the graph from [5].…”

“…Upon close examination, one can see that the proposition follows from the proof of lemma 3 in [5]. To show this, first recall the definition of the graph from [5]. Their final graph, which we denote by G, is constructed out of boundary-anchored geodesics in the same way asΓ (except for the final step where the two sets of boundary vertices are merged); however, the set of geodesics is different.…”

“…Our approach to identifyingà is to reformulate the question as a problem on a graph using a construction that is a variation on the one presented in section 3 of [5]. Without loss of generality, suppose that the A i are numbered according to the order in which they appear going counter-clockwise along ∂X and let A i = [a i , b i ], again with respect to counterclockwise ordering.…”

“…Moreover, the proof of this lemma further establishes that the minimal cut, call it C * I , actually corresponds to the Ryu-Takayanagi surfaceà I (or possibly another equivalent surface with the same length if the minimal surface is not unique). Now, if the two sets of geodesics that are used to define our graphΓ and the graph G from [5] had been the same, then the proof would be complete since we would have that…”

“…Following previous inspiring and precise statements of a connection between the Ryu-Takayanagi conjecture and max-flow/min-cut [4,5], we first devise a constructive algorithm that reduces the problem of determining the minimal-area surface in the context of AdS 3 /CFT 2 to solving max-flow/min-cut on a graph. Crucially, the latter problem can be solved in polynomial time.…”