Southwest Quantum Information and Technology

The holographic entropy cone

Presenting Author: Sepehr Nezami, Stanford Institute For Theoretical Physics
Contributing Author(s): Ning Bao, Hires Ooguri, Bogdan Stoica, James Sully, Michael Walter

After discovery of the AdS/CFT correspondence, it was known that some conformal field theories (CFT) are dual to a gravitational theory in a higher dimension. Moreover, Ryu and Takayanagi showed that the entropy of a region of a CFT state is given by the area of a minimal surface in the dual gravitational theory. We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with 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 subsystems. We also find a new infinite 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.

Read this article online: http://arxiv.org/abs/1505.07839

(Session 5 : Thursday from 5:00 - 7:00 pm)