The holographic entropy cone

By Ning Bao, Sepehr Nezami, Hirosi Ooguri, Bogdan Stoica, J. Sully, Michael Walter. 2015

223 citations recorded by the source corpus.

A bstractWe 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.

Subfields: Applied Mathematics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/9f925e4b0b96067a953356225ea8d73b9bd65a84

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. D. Harlow
2. M. Headrick
3. T. Takayanagi
4. T. Nishioka
5. M. Nozaki
6. Thomas Hartman

In Geometry and Topology

1. M. Headrick
2. T. Takayanagi
3. D. Harlow
4. Michael H. Freedman
5. Patrick M. Hayden
6. R. Myers
7. V. Hubeny
8. M. Rangamani

In Nuclear and High Energy Physics

1. D. Harlow
2. T. Takayanagi
3. M. Headrick
4. D. Stanford
5. Thomas Hartman
6. R. Myers
7. Patrick M. Hayden
8. Michael H. Freedman

Citations in the corpus

• The Ryu–Takayanagi Formula from Quantum Error Correction (2016, background): “Since we know that the thread prescription is equivalent to the area term of the RT formula, once we consider four regions the entropies will obey inequalities such as the monogamy of mutual information that are not actually true for general quantum states [79,80], so at that point we will start see”
• Bit Threads and Holographic Entanglement (2016, background): “Our main interest here is in the fact that, in holographic theories, the tripartite information is always non-positive, I3(A : B : C) ≤ 0 (2.22) a property called16 “monogamy of mutual information” (MMI) [8,20].”
• Holographic duality from random tensor networks (2016, background): “2) is that any entropy inequality that is valid for arbitrary quantum states, or even just for stabilizer states [54, 55], is also valid for the Ryu-Takayanagi entropy formula, thereby establishing a conjecture from [22].”
• Space from Hilbert Space: Recovering Geometry from Bulk Entanglement (2016, background): “As such, they form a superset of area-law states which also satisfy the holographic inequalities [50].”
• Entanglement of purification through holographic duality (2017, background): “A large class of such entropic inequalities for holographic states has been found in [15].”