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Relative entropy and holography

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Abstract

Relative entropy between two states in the same Hilbert space is a fundamental statistical measure of the distance between these states. Relative entropy is always positive and increasing with the system size. Interestingly, for two states which are infinitesimally different to each other, vanishing of relative entropy gives a powerful equation ΔS = ΔH for the first order variation of the entanglement entropy ΔS and the expectation value of the modular Hamiltonian ΔH. We evaluate relative entropy between the vacuum and other states for spherical regions in the AdS/CFT framework. We check that the relevant equations and inequalities hold for a large class of states, giving a strong support to the holographic entropy formula. We elaborate on potential uses of the equation ΔS = ΔH for vacuum state tomography and obtain modified versions of the Bekenstein bound.

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Authors and Affiliations

  1. Centro Atómico Bariloche, 8400, S.C. de Bariloche, Río Negro, Argentina

    David D. Blanco & Horacio Casini

  2. Department of Physics, Harvard University, Cambridge, MA, 02138, U.S.A.

    Ling-Yan Hung

  3. Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada

    Robert C. Myers

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  1. David D. Blanco
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  2. Horacio Casini
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Correspondence to Ling-Yan Hung.

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Blanco, D.D., Casini, H., Hung, LY. et al. Relative entropy and holography. J. High Energ. Phys. 2013, 60 (2013). https://doi.org/10.1007/JHEP08(2013)060

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