Entanglement between two gravitating universes

TL;DR

The paper analyzes entanglement between two disjoint gravitating universes using the replica trick in JT gravity coupled to a CFT. When both universes are gravitating, a fully connected $M_{2n}$ wormhole dominates and, after a $\mathbb{Z}_n$ quotient, reduces to a cylinder described by a swap wormhole in the high-entanglement limit, realizing an ER=EPR bridge that restores unitarity. The resulting entropy takes a generalized form on the glued $A/B$ spacetime, highlighting a tidal-island–like mechanism that ties entanglement to geometry. These results illuminate how entanglement can drive nontrivial spacetime connections and refine island-type prescriptions by introducing a bulk geometry that spans the two gravitating sectors.

Abstract

We study two disjoint universes in an entangled pure state. When only one universe contains gravity, the path integral for the $n^{\text{th}}$ 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 $\mathbb{Z}_n$ 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.

Figures (11)

• Figure 1: We have two universes $A$ and $B$ with semiclassical Euclidean disk solutions.
• Figure 2: Possible gravitational configurations connecting copies of the universes $A$ and $B$ in the $n=2$ Rényi entropy \ref{['eq:renyithiscase']}. (Top left) Type I configuration where all copies are disconnected. (Top right) Type II$_{A}$ configuration where all copies of the universe $A$ are connected by a replica wormhole. (Bottom left) Type III configuration where all copies of $A$ are connected by a replica wormhole, and all copies of $B$ are connected by another replica wormhole, but these two wormholes are not connected. (Bottom right) Type IV configuration where all copies are connected by a single wormhole.
• Figure 3: The $n$-sheeted cylinder manifold $\Sigma^{\text{cyl}}_n$ with operator insertions appearing in \ref{['eq:cylindercorrl']}. Going through the cut takes us to a different sheet, on which the particular operator insertions are different. The cut here is in grey compared to Fig. \ref{['fig:replica']} because here we have only deformed the manifold $M_{2n}$ slightly, as opposed to taking a quotient of the topology by a $\mathbb{Z}_n$ replica symmetry. Indeed, with the operator insertions, this path integral on its own is not actually replica symmetric, and one of our main aims in Sec. \ref{['sec:effective']} is to start with the CFT path integral on this branched manifold and manipulate it into a nicer form more amenable to analytic continuation in $n$.
• Figure 4: Two moduli of the cylinder wormhole $b$ and $\tau$. The twist parameter $\tau$ rotates universe $B$, and the circumference $b$ measures the width of the wormhole. The twist changes the distance between $\psi_{j_{k}} (\infty_{A})$ in universe $A$ and $\psi_{j_{k}} (0_{B})$ in universe $B$ in the correlator \ref{['eq:cylindercorrl']}. This distance is minimized when $\tau = \pi$. We can also think of the renormalized length $\ell$ between two boundary points, which becomes small when the circumference $b$ is large.
• Figure 5: Left: The short length or large circumference limit of the wormhole connecting the universes $A$ and $B$ where the cylinder has been drawn as an annulus and we have already set $\tau = \pi$ to rotate the $A$ boundary relative to the $B$ boundary. The black dots represent operator insertions on the cylinder wormhole, and we have drawn the cut in its extremized location between $\psi_{i_k}$ and $\psi_{i_{k+1}}$. Right: When $\ell \rightarrow 0$ or $b\to\infty$, the long Euclidean evolutions on the annulus may be replaced with CFT vacuum projectors. We have exaggerated the Euclidean evolution on the right.