Linearity of holographic entanglement entropy

Received: July Linearity of holographic entanglement entropy Ahmed Almheiri 0 1 3 Xi Dong 0 1 2 Brian Swingle 0 1 3 Open Access 0 1 c The Authors. 0 1 0 nd that this fails once 1 Stanford University , Stanford, CA 94305 , U.S.A 2 School of Natural Sciences, Institute for Advanced Study 3 Stanford Institute for Theoretical Physics, Department of Physics We consider the question of whether the leading contribution to the entanglement entropy in holographic CFTs is truly given by the expectation value of a linear operator as is suggested by the Ryu-Takayanagi formula. We investigate this property by computing the entanglement entropy, via the replica trick, in states dual to superpositions of macroscopically distinct geometries and nd it consistent with evaluating the expectation value of the area operator within such states. However, we the number of semi-classical states in the superposition grows exponentially in the central charge of the CFT. Moreover, in certain such scenarios we nd that the choice of surface on which to evaluate the area operator depends on the density matrix of the entire CFT. This nonlinearity is enforced in the bulk via the homology prescription of Ryu-Takayanagi. We thus conclude that the homology constraint is not a linear property in the CFT. We also discuss the existence of `entropy operators' in general systems with a large number of degrees of freedom. AdS-CFT Correspondence; Black Holes in String Theory; Gauge-gravity cor- - Entanglement entropy and the replica trick A single interval example Superpositions of one-sided AdS spacetimes Superpositions of eternal black holes Linearity vs. homology A failure of linearity: homology The source of the homology constraint in the CFT Entanglement entropy for superpositions of semi-classical states Entropy operators more generally N copies of a qubit N copies of a free eld theory Di erent sets of states Considerations and future directions Conditions for a semi-classical spacetime Quantum error correction and superpositions A nonlinearity for single sided pure states Connections to one-shot information theory Tensor networks for superpositions Comment on (non)linearity of Renyi entropies A Entanglement entropy for a semi-classical superposition 1 Introduction The area operator of the Ryu-Takayanagi proposal A gauge invariant area operator The boundary support of the area operator The linearity of the area operator The area operator on superpositions | A prediction How to compute entanglement entropy in 1 + 1 CFTs usually equated. Entropy is not a linear operator while area is, yet in gravity these two quantities are This was rst observed in the context of black hole thermodynamics where it was shown that the entropy of a black hole is given by the expectation value of the area operator evaluated on its event horizon [1]. This operator is a nonlinear functional of the canonical variables of quantum gravity (the metric and conjugate momentum) and is understood to be a linear operator which maps states to states. Given that this entropy is a coarse-grained thermodynamic quantity, it seems plausible that it can be represented by a linear operator very much in the same way that the entropy of a gas can be represented by its energy. One should probably expect this property in systems with a thermodynamic limit and which are known to thermalize. A more paradoxical relationship between entropy and area arises in the context of the AdS/CFT correspondence. This correspondence is a duality between string theories living in d + 1-dimensional asymptotically Anti de Sitter (AdS) space and certain d-dimensional conformal eld theories (CFTs) which can be thought of as living on the boundary of AdS [2]. One way in which these two descriptions are connected is via the identi cation of the central charge of the CFT with the ratio of the AdS length to the Planck length to some positive power, c of a certain class of theories of quantum gravity in asymptotically AdS spacetimes in terms of a certain class of CFTs. An outcome of this duality is that the strong coupling and c ! 1 limit of the CFT is described on the AdS (bulk) side by classical gravity with a gravitational constant GN remarkably simple, albeit confusing, formula for the entanglement entropy of any region of the CFT was proposed [3]. It was suggested that, in static situations, the entanglement entropy of a subregion R of the CFT is given by the area of the minimal area bulk surface SR = We shall refer to this henceforth as the RT formula. This formula was proven in [4] under certain reasonable assumptions including the extension of the replica symmetry into the dominant bulk solution. It was also extended to the time dependent case in [5] where the minimal surface generalizes to a spacelike extremal surface. Another proposal for the time dependent case was presented in [6]; their prescription was to nd the minimum area X on every possible spatial slice containing the int (...truncated)