Quantum corrections to holographic entanglement entropy

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Abstract

We consider entanglement entropy in quantum field theories with a gravity dual. In the gravity description, the leading order contribution comes from the area of a minimal surface, as proposed by Ryu-Takayanagi. Here we describe the one loop correction to this formula. The minimal surface divides the bulk into two regions. The bulk loop correction is essentially given by the bulk entanglement entropy between these two bulk regions. We perform some simple checks of this proposal.

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Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy

Robert Wald, Vivek Iyer (1994)

We consider a general, classical theory of gravity with arbitrary matter fields in \(n\) dimensions, arising from a diffeomorphism invariant Lagrangian, \(\bL\). We first show that \(\bL\) always can be written in a ``manifestly covariant" form. We then show that the symplectic potential current \((n-1)\)-form, \(\th\), and the symplectic current \((n-1)\)-form, \(\om\), for the theory always can be globally defined in a covariant manner. Associated with any infinitesimal diffeomorphism is a Noether current \((n-1)\)-form, \(\bJ\), and corresponding Noether charge \((n-2)\)-form, \(\bQ\). We derive a general ``decomposition formula" for \(\bQ\). Using this formula for the Noether charge, we prove that the first law of black hole mechanics holds for arbitrary perturbations of a stationary black hole. (For higher derivative theories, previous arguments had established this law only for stationary perturbations.) Finally, we propose a local, geometrical prescription for the entropy, \(S_{dyn}\), of a dynamical black hole. This prescription agrees with the Noether charge formula for stationary black holes and their perturbations, and is independent of all ambiguities associated with the choices of \(\bL\), \(\th\), and \(\bQ\). However, the issue of whether this dynamical entropy in general obeys a ``second law" of black hole mechanics remains open. In an appendix, we apply some of our results to theories with a nondynamical metric and also briefly develop the theory of stress-energy pseudotensors.

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Holographic Derivation of Entanglement Entropy from AdS/CFT

, (2010)

A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from AdS/CFT correspondence. We argue that the entanglement entropy in d+1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS_{d+2}, analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal perfectly reproduces the correct entanglement entropy in 2D CFT when applied to AdS_3. We also compare the entropy computed in AdS_5 \times S^5 with that of the free N=4 super Yang-Mills.

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Black Hole Entropy in Canonical Quantum Gravity and Superstring Theory

J. Uglum, L. Susskind (1994)

In this paper the entropy of an eternal Schwarzschild black hole is studied in the limit of infinite black hole mass. The problem is addressed from the point of view of both canonical quantum gravity and superstring theory. The entropy per unit area of a free scalar field propagating in a fixed black hole background is shown to be quadratically divergent near the horizon. It is shown that such quantum corrections to the entropy per unit area are equivalent to the quantum corrections to the gravitational coupling. Unlike field theory, superstring theory provides a set of identifiable configurations which give rise to the classical contribution to the entropy per unit area. These configurations can be understood as open superstrings with both ends attached to the horizon. The entropy per unit area is shown to be finite to all orders in superstring perturbation theory. The importance of these conclusions to the resolution of the problem of black hole information loss is reiterated.