Deriving covariant holographic entanglement

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.

Entanglement entropy in flat holography

“…A covariant version was proposed by Hubeny, Rangamani and Takayanagi (HRT) [59]. Using AdS/CFT for Einstein gravity, RT and HRT proposal have been proved by [60][61][62][63][64]. It is interesting to ask if the connection between spacetime structure in the bulk and entanglement in the boundary still exist beyond the context of AdS/CFT, and if so, how it works for non-AlAdS spacetimes.…”

“…The second approach is to directly propose a prescription in the bulk, and check its consistency [70,71]. The third approach, which we will advocated in the current paper, is to derive an analog of RT proposal using the dictionary of holography, along the lines of [60][61][62][63][64]. In [72,73], holographic entanglement entropy in Warped AdS 3 spacetime was derived by generalizing the gravitational entropy [63] and Rindler method [60,74], respectively.…”

“…• The second route is to extend replica symmetry to the bulk along the lines of [63,64]. The field theory generator k t has a bulk extension k bulk t via the holographic dictionary.…”

Discrete gravity on random tensor network and holographic Rényi entropy

“…However the tensor networks designed in [24,25] only consider the geometry of a spatial slice in AdS, without any input about time evolution. In the continuum AdS/CFT context, the correct entanglement spectrum are obtained by considering the spacetime geometry, and taking into account the bulk dynamics given by the Einstein equation [11,12,14,15], while [24,25] don't have any dynamical input. In this work, we take the above idea as a hint to improve the random tensor network approach, in which we resolve the issue.…”

Time-independent wormholes

“…This includes the (leading order) holographic mutual information defined by the Ryu-Takayanagi (RT) [11,12] or the covariant Hubeny-RangamaniTakayanagi (HRT) [13] prescriptions. While -as will be discussed in section 4 -the derivations of [14] and [15] need not apply to our spacetime, it is nevertheless of interest to investigate what these prescriptions would predict. In particular, we will see thatdespite the instability noted above -in a sense these mutual informations (and indeed the entropies of all boundary regions) appear to already be thermalized at any finite t.…”