Islands and mixed states in closed universes

TL;DR

This work extends the island paradigm to closed universes by analyzing a three-boundary AdS$_3$ braneworld where a gravitating brane (the closed universe) is entangled with two non-gravitating CFTs. By comparing RT surfaces in a multiboundary wormhole, the authors show that the mixed-state entropy on the brane is bounded above by half of the coarse-grained brane entropy and by the island boundary contribution, with transitions governed by the minimal RT surface $\mathcal{W}$ that ends on the brane. The analysis provides concrete expressions for the brane entropy in different entanglement regimes and demonstrates how island physics encodes the semi-classical mixing on the brane into a microscopic mixed state. The results illuminate how information about a closed universe can be recoverable from external quantum systems and point to generalizations in higher dimensions and more complex reference-system couplings.

Abstract

We investigate the appearance of islands when a closed universe with gravity is entangled with a non-gravitating quantum system. We use braneworlds in three-dimensional multiboundary wormhole geometries as a model to explore what happens when the non-gravitating system has several components. The braneworld can be either completely contained in the entanglement wedge of one of the non-gravitating systems or split between them. In the former case, entanglement with the other system leads to a mixed state in the closed universe, unlike in simpler setups with a single quantum system, where the closed universe was necessarily in a pure state. We show that the entropy of this mixed state is bounded by half of the coarse-grained entropy of the effective theory on the braneworld.

Figures (5)

• Figure 1: A constant-time slice of the geometry with an end of the world brane in Poincaré-AdS, showing the RT surface for a region $x \in (-L,0)$ on the boundary
• Figure 2: The pair of pants geometry with an end of the world brane (red) in one asymptotic region. On the left is a cartoon of the geometry of the $t=0$ surface, and on the right is its description as a quotient of the Poincaré disc model. In the right picture the central region bounded by the blue and orange geodesics is a fundamental region for the identification. The geodesic $\mathcal{W}\equiv\mathcal{W}'\cup\mathcal{W}"$, which is the minimal geodesic anchored on the end of the world brane running in between the two asymptotic region, is shown in green. The horizon $H_3$ is similarly defined as $H_3\equiv H_3'\cup H_3"$.
• Figure 3: In the large $\ell_i$ limit, the horizons are locally identified. There are two cases: if one length is larger than the sum of the other two, $\ell_i > \ell_j + \ell_k$, we have an "eyeglass" picture, where the whole of the two short horizons are identified with the long one, and the remaining parts of the long horizon are identified with each other. Otherwise, each horizon has a portion which is identified with each of the others.
• Figure 4: The regions where $\ell_{\mathcal{W}} < \ell_2$ for $\ell_2=\ell_3$ for various values of $T$
• Figure 5: For a brane in an eternal black hole spacetime, when we consider a subregion of the CFT on the boundary, the RT surface is either outside the horizon or crosses the horizon and ends on the brane. For large black holes, the surface outside the horizon has a portion which lies along the horizon, whose area can be interpreted as entropy of the mixed state on the brane.