Noncommutative Geometry Seminar

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.