Entanglement between two disjoint universes

TL;DR

The paper studies entanglement between a non-gravitating universe A and a gravitating universe B using the replica trick with Euclidean wormholes to derive an island in JT gravity. It shows that the island coincides with a causal shadow in B and that, at high entanglement temperature, the island enforces a Page-like bound while the entanglement wedge of A grows to eventually reconstruct the entire bulk. The analysis spans sphere and disk topologies, demonstrates robustness against classical correlations, and provides a holographic BTZ description via RT surfaces. Collectively, the work illustrates how semiclassical gravity encodes monogamy of entanglement and enables reconstruction of gravitational regions from a distant quantum system, highlighting a general mechanism for island formation beyond black holes.

Abstract

We use the replica method to compute the entanglement entropy of a universe without gravity entangled in a thermofield-double-like state with a disjoint gravitating universe. Including wormholes between replicas of the latter gives an entropy functional which includes an "island" on the gravitating universe. We solve the back-reaction equations when the cosmological constant is negative to show that this island coincides with a causal shadow region that is created by the entanglement in the gravitating geometry. At high entanglement temperatures, the island contribution to the entropy functional leads to a bound on entanglement entropy, analogous to the Page behavior of evaporating black holes. We demonstrate that the entanglement wedge of the non-gravitating universe grows with the entanglement temperature until, eventually, the gravitating universe can be entirely reconstructed from the non-gravitating one.

Figures (7)

• Figure 1: We consider the entanglement between two disjoint universes, $A$ and $B$. Both universes have propagating matter, but only $B$ gravitates. The universes may be closed and bounded as illustrated, or non-compact.
• Figure 2: Left: The replica manifold $M_{\text{conn}}$ for $n=3$, with operator insertions which differ on the same sheet. The blue wiggly line is the cut $C_x$ with length $2\pi x$, and the spheres are glued cyclically along it. Right: the replica manifold $M_{\text{conn}}$, with operator insertions that are the same on a given sheet. The spheres are again glued cyclically, this time along the red wiggly line $\overline{C_x}$, which is a cut of length $2\pi(1-x)$. The path integrals on these two manifolds with the specified operator insertions are equivalent, as proven in Appendix \ref{['sec:identity']}.
• Figure 3: The Penrose diagram of the backreacted black hole with the dilaton profile \ref{['eq:totaldil']}. Near asymptotic boundaries $\mu \rightarrow \frac{\pi}{2}$, it looks like an eternal black hole \ref{['eq:ordinarybh']} with a fixed temperature $1/\beta_{BH}$. The shaded region is the causal shadow region, which is not causally accessible from both of the asymptotic boundaries. As we increase the entanglement temperature $1/\beta$, the size of the causal shadow region increases, and the event horizons approach the boundaries. The blue line in the $\tau=0$ slice is the island of the black hole.
• Figure 4: Plots of the dilaton profile \ref{['eq:totaldil']} for $b=5$ (left) and $b=1000$ (right). Larger $b$ corresponds to higher temperature. The dilaton is minimized at the horizon, and its minimum value corresponds to the classical entropy of the black hole.
• Figure 5: A plot of the two candidate entropies for $S_A$ as a function of the CFT temperature $T = 1/\beta$, with $\phi_0 = 1000$, $\phi_b = c = 10$, and $L = G_N =1$. The blue curve is the thermal entropy of CFT fields on universe $A$, and corresponds to dominance of the disconnected saddle in the replica trick computation of the entropy. The red curve is the generalized entropy of a complement island on universe $B$, which corresponds to dominance of the fully connected replica wormhole. The Page curve in AdS (the entropy $S_A$) is computed by taking the minimum of these two curves. We see that the Page transition occurs around $T = 8$.