Journal of High Energy Physics

, 2015:130 | Cite as

The holographic entropy cone

  • Ning Bao
  • Sepehr Nezami
  • Hirosi Ooguri
  • Bogdan Stoica
  • James Sully
  • Michael WalterEmail author
Open Access
Regular Article - Theoretical Physics


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.


Gauge-gravity correspondence AdS-CFT Correspondence Black Holes in String Theory 2D Gravity 


Open Access

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Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Ning Bao
    • 1
    • 2
  • Sepehr Nezami
    • 3
  • Hirosi Ooguri
    • 2
    • 4
  • Bogdan Stoica
    • 2
  • James Sully
    • 5
  • Michael Walter
    • 3
    Email author
  1. 1.Institute for Quantum Information and Matter, California Institute of TechnologyPasadenaUnited States
  2. 2.Walter Burke Institute for Theoretical Physics, California Institute of TechnologyPasadenaUnited States
  3. 3.Stanford Institute for Theoretical PhysicsStanford UniversityStanfordUnited States
  4. 4.Kavli Institute for the Physics and Mathematics of the UniverseUniversity of TokyoKashiwaJapan
  5. 5.Theory Group, SLAC National Accelerator LaboratoryStanford UniversityMenlo ParkUnited States

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