The holographic entropy cone
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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.
KeywordsGauge-gravity correspondence AdS-CFT Correspondence Black Holes in String Theory 2D Gravity
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- M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [Int. J. Mod. Phys. D 19 (2010) 2429] [arXiv:1005.3035] [INSPIRE].
- N. Pippenger, What are the laws of information theory?, Spec. Prob. Comm. Comp. Conf., Palo Alto U.S.A. (1986).Google Scholar
- F. Matús, Infinitely many information inequalities, in Proc. Int. Symp. Inf. Theor. (ISIT), (2007), pg. 41.Google Scholar
- N. Linden and A. Winter, A new inequality for the von Neumann entropy, Commun. Math. Phys. 259 (2005) 129 [quant-ph/0406162].
- N. Linden, F. Matús, M.B. Ruskai and A. Winter, The quantum entropy cone of stabiliser states, in Proc. 8th TQC Guelph, LIPICS, vol. 22, (2013), pg. 270 [arXiv:1302.5453].
- C. Majenz, Constraints on multipartite quantum entropies, Master’s thesis, University of Freiburg, Freiburg Germany (2014).Google Scholar
- B. Ibinson, Quantum information and entropy, Ph.D. thesis, University of Bristol, Bristol U.K. (2006).Google Scholar
- M. Walter, Multipartite quantum states and their marginals, Ph.D. thesis, ETH Zurich, Zurich Switzerland (2014) [arXiv:1410.6820].
- A. Ingleton, Representation of matroids, in Combinatorial mathematics and its applications, D. Welsh ed., Academic Press, U.S.A. (1971), pg. 149.Google Scholar
- R. Dougherty, C. Freiling and K. Zeger, Linear rank inequalities for five or more variables, arXiv:0910.0284.
- E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Transformation groups for soliton equations, in Proceedings of RIMS Symposium on Non-linear Integrable Systems (Kyoto, 1981), M. Jimbo and T. Miwa eds., World Scientific, Singapore (1983), pg. 39.Google Scholar
- F. Luo, Geodesic length functions and Teichmüller spaces, J. Diff. Geom. 48 (1998) 275 [math.GT/9801024].