Entanglement between two gravitating universes

Abstract

We study two disjoint universes in an entangled pure state. When only one universe contains gravity, the path integral for the n th Rényi entropy includes a wormhole between the n copies of the gravitating universe, leading to a standard “island formula” for entanglement entropy consistent with unitarity of quantum information. When both universes contain gravity, gravitational corrections to this configuration lead to a violation of unitarity. However, the path integral is now dominated by a novel wormhole with 2n boundaries connecting replica copies of both universes. The analytic continuation of this contribution involves a quotient by Ζ n replica symmetry, giving a cylinder connecting the two universes. When entanglement is large, this configuration has an effective description as a “swap wormhole”, a geometry in which the boundaries of the two universes are glued together by a “swaperator”. This description allows precise computation of a generalized entropy-like formula for entanglement entropy. The quantum extremal surface computing the entropy lives on the Lorentzian continuation of the cylinder/swap wormhole, which has a connected Cauchy slice stretching between the universes – a realization of the ER=EPR idea. The new wormhole restores unitarity of quantum information.

Entanglement between two evaporating black holes

“…In this paper, we generalize this entropy calculation, and discuss more involved setups where the black hole mass in the universe A is different from the one in the universe B, to further understand the detailed properties of the formula (1.3). In such situations, there are several possibilities for the glued geometry A/B, which are indistinguishable in the previous setup [54]. Our results show that the dominant glued geometry A/B, which contributes the entropy formula (1.3) is given by the one which minimizes the total black hole entropy in A/B.…”

“…However, if we consider a bath which is also gravitating, then the situation drastically changes and becomes more complex. Such setups are studied in [40], see also [48,53,54,62]. However, there are subtle points in defining the von Neumann entropy in the presence of dynamical gravity.…”

“…One way to avoid these subtleties is to prepare two disjoint gravitating universes A and B, then make them entangled [54] 1 . In this setup, one can take the entangled state to be the thermo-field double state on the two universes…”

“…We will study this state in detail in next section 2. In computing the entanglement entropy of the state on the universe A (or the universe B) by using the replica trick, the dominant replica wormhole contribution is the one connecting all the gravitating universe's replicas when the entanglement between the two universes is sufficiently large β → 0 [54]. Such a contribution leads us to the formula,…”

“…In the previous paper [54], only the simplest setup where the universes A, B have eternal black holes with a same black hole mass (or equivalently a same black hole entropy) was < l a t e x i t s h a 1 _ b a s e 6 4 = " N 0 O 1 o 1 Z c I P + u R d j I 4 J z p j 1 + 4 y 7 c = " > A…”