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arXiv:1404.0602 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Entanglement entropy and negative energy in two dimensions

Eugenio Bianchi A, Matteo Smerlak

Published Aug 27, 2014• 67 citations
arXivHTMLPDF

Abstract

It is well known that quantum effects can produce negative energy densities, though for limited times. Here we show in the context of two-dimensional conformal field theory that such negative energy densities are present in any nontrivial conformal vacuum and can be interpreted in terms of the entanglement structure of such states. We derive an exact identity relating the outgoing energy flux and the entanglement entropy in the in-vacuum. When applied to two-dimensional models of black hole evaporation, this identity implies that unitarity is incompatible with monotonic mass loss.

DOI
10.1103/physrevd.90.041904

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It is well known that quantum effects can produce negative energy densities, though for limited times. Here we show in the context of two-dimensional conformal field theory that such negative energy densities are present in any nontrivial conformal vacuum and can be interpreted in terms of the entanglement structure of such states. We derive an exact identity relating the outgoing energy flux and the entanglement entropy in the in-vacuum. When applied to two-dimensional models of black hole evaporation, this identity implies that unitarity is incompatible with monotonic mass loss.
Using the AdS/CFT correspondence we derive a formula for the entanglement entropy of the anti-de Sitter black hole in two spacetime dimensions. The leading term in the large black hole mass expansion of our formula reproduces exactly the Bekenstein–Hawking entropy SBH, whereas the subleading term behaves as lnSBH. This subleading term has the universal form typical for the entanglement entropy of physical systems described by effective conformal fields theories (e.g. one-dimensional statistical models at the critical point). The well-known form of the entanglement entropy for a two-dimensional conformal field theory is obtained as analytic continuation of our result and is related with the entanglement entropy of a black hole with negative mass.
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We explore entanglement negativity, a measure of the distillable entanglement contained in a quantum state, in relativistic field theories in various dimensions. We first give a general overview of negativity and its properties and then explain a well known result relating (logarithmic) negativity of pure quantum states to the Rényi entropy (at index 1/2), by exploiting the simple features of entanglement in thermal states. In particular, we show that the negativity of the thermofield double state is given by the free energy difference of the system at temperature T and 2 T respectively. We then use this result to compute the negativity in the vacuum state of conformal field theories in various dimensions, utilizing results that have been derived for free and holographic CFTs in the literature. We also comment upon general lessons to be learnt about negativity in holographic field theories.
The trace over the degrees of freedom located in a subset of the space transforms the vacuum state into a mixed density matrix with nonzero entropy. This is usually called entanglement entropy, and it is known to be divergent in quantum field theory (QFT). However, it is possible to define a finite quantity F(A,B) for two given different subsets A and B which measures the degree of entanglement between their respective degrees of freedom. We show that the function F(A,B) is severely constrained by the Poincaré symmetry and the mathematical properties of the entropy. In particular, for one component sets in two-dimensional conformal field theories its general form is completely determined. Moreover, it allows to prove an alternative entropic version of the c-theorem for (1+1)-dimensional QFT. We propose this well-defined quantity as the meaningfull entanglement entropy and comment on possible applications in QFT and the black hole evaporation problem.
The quantum information content of Hawking radiation holds the key to understanding black-hole evaporation and the fate of unitarity. Motivated by recent advances in cold-atom experiments, we develop a lattice-regularization approach aimed at simulating the coarse-grained entanglement scaling of a quantum field in a 1+1D analogue black-hole background. We provide the first concrete demonstration that logarithmic negativity -- an entanglement monotone that typically exhibits a UV-divergent log-scaling for the conformal vacuum -- acquires a UV-finite volume term from the nonlocal correlations seeded by Hawking radiation. We show that this volume term encodes the number density as well as the spatial distribution of entangled Hawking pairs along the black-hole interior and exterior. We highlight its prospective detection in currently realizable experiments and its implications beyond the analogue paradigm, in particular for black-hole thermodynamics.