Deriving covariant holographic entanglement

Published for SISSA by Springer Received: September 2, 2016 Accepted: October 4, 2016 Published: November 7, 2016 Deriving covariant holographic entanglement a School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, U.S.A. b Jadwin Hall, Princeton University, Princeton, NJ 08544, U.S.A. c Center for Quantum Mathematics and Physics (QMAP), Department of Physics, University of California, Davis, CA 95616 U.S.A. E-mail: , , Abstract: We provide a gravitational argument in favour of the covariant holographic entanglement entropy proposal. In general time-dependent states, the proposal asserts that the entanglement entropy of a region in the boundary field theory is given by a quarter of the area of a bulk extremal surface in Planck units. The main element of our discussion is an implementation of an appropriate Schwinger-Keldysh contour to obtain the reduced density matrix (and its powers) of a given region, as is relevant for the replica construction. We map this contour into the bulk gravitational theory, and argue that the saddle point solutions of these replica geometries lead to a consistent prescription for computing the field theory Rényi entropies. In the limiting case where the replica index is taken to unity, a local analysis suffices to show that these saddles lead to the extremal surfaces of interest. We also comment on various properties of holographic entanglement that follow from this construction. Keywords: AdS-CFT Correspondence, Classical Theories of Gravity, Gauge-gravity correspondence ArXiv ePrint: 1607.07506 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP11(2016)028 JHEP11(2016)028 Xi Dong,a Aitor Lewkowyczb and Mukund Rangamanic Contents 1 2 Field theory construction 2.1 The time-dependent QFT wavefunctional 2.2 The reduced density matrix 2.3 Products of reduced density matrix and Rényi entropies 6 7 7 11 3 Gravitational construction 3.1 Review: time reflection symmetric case 3.2 Covariant generalization to time-dependent situations 3.2.1 Kinematics: setting up the bulk construction 3.2.2 Dynamics: equations of motion and extremal surfaces 13 13 18 20 24 4 Discussion 26 A Bulk evaluation of the Rényi entropy A.1 Evaluation of boundary term A.2 Example of bulk integral 32 33 34 1 Introduction One of the intriguing aspects of the holographic AdS/CFT correspondence is the geometrization of quantum entanglement. In a local QFT the amount of correlation between degrees of freedom confined in a spatial region, and those outside, is measured by the entanglement entropy. While simple to state, this quantity is notoriously hard to compute in all but a handful of circumstances. At a technical level its computation requires determining the logarithm of a state-dependent operator (see below), which is challenging in an interacting QFT. Given this situation, it is rather remarkable that one has a rather simple way to compute entanglement in holographic field theories thanks to the AdS/CFT correspondence. Specifically, Ryu and Takayanagi (RT) [1, 2] argued, drawing analogy with the behaviour of black hole entropy in gravitational theories, that the entanglement entropy ascribable to a region A is given by solving a classical geometric problem in AdS. One is instructed to find a minimal area surface anchored on the boundary ∂A of the region of interest; its area in Planck units measures the entanglement entropy SA . The RT prescription was originally given for static states; this extends trivially to more generally to states at a moment of time reflection symmetry. However, the concept of entanglement being quite –1– JHEP11(2016)028 1 Introduction 1 It should be noted that general covariance by itself is not strong enough [3]. For e.g., one can use causal structures to motivate a different construction leading to the causal holographic information [4]. See also [5] for a discussion of the merits of general covariance as a guiding principle in a closely related context. –2– JHEP11(2016)028 fundamental is not limited to such situations alone. Indeed the notion of entanglement entropy makes sense even when the state in question involves non-trivial temporal evolution. The Hubeny-Rangamani-Takayanagi (HRT) proposal [3] mitigates this lacuna by arguing that the correct extension of the RT prescription in time-dependent situations involves consideration of codimension-2 extremal surfaces. The primary intuition behind the HRT proposal was to ask what is the correct covariant generalization of the RT prescription. As we know from other aspects of gravitational physics, the principle of general covariance is a strong guiding principle, which in conjunction with other physical requirements serves to almost always zero in on the (oftentimes unique) dynamical construction. Demanding that the RT prescription admit a covariant upgrade, along with agreement in a common domain of applicability, led HRT to the extremal surface prescription.1 Whilst these prescriptions for computing entanglement entropy in the holographic context are extremely simple, one would like to derive them from first principles using nothing but the basic entries of the AdS/CFT dictionary. This has been achieved for the RT prescription explicitly by Lewkowycz and Maldacena (LM) [6] who mapped the replica construction usually employed to compute entanglement entropy in quantum field theories to the Euclidean quantum gravity path integral. In this context it is worth mentioning [7] who argued for the RT prescription in the case of spherical regions in the vacuum of a CFT (see also [8] for an initial attempt at a proof). The argument forwarded by LM roughly proceeds as follows: given a density matrix ρA a measure of entanglement is provided by the von Neumann entropy SA = − Tr(ρA log ρA ). In practice, owing to the technical complexities of taking the logarithm of an operator, one (q) 1 instead computes the Rényi entropies SA = 1−q log Tr(ρA q ) for q ∈ Z+ [9]. Analytically continuing these Rényi entropies away from integral Rényi index q, one obtains the von Neumann entropy in the limit q → 1. When the density matrix is at a moment of time-reflection symmetry (or more simply just time-translationally invariant) one can employ Euclidean path integral techniques. One formally writes ρA = e−2πKA which defines the modular Hamiltonian KA and views the computation of the q th Rényi entropy as an evolution around an ‘Euclidean replica circle’ parameterized by τ with τ ∼ τ + 2πq. This is entirely analogous to the computation of the canonical partition function by evolving along the Euclidean thermal circle for a period set by the inverse temperature before taking the trace. This analogy is in fact exact in the case of the spherical regions A in the vacuum state of a CFT [7], for in this case the reduced density matrix is unitarily equivalent to the thermal density matrix on hyperbolic space (by a conformal mapping). The LM construc (...truncated)