Bra-ket wormholes in gravitationally prepared states

TL;DR

The paper investigates how gravitational path integrals in 2D JT gravity coupled to a CFT can prepare boundary states in flat space, focusing on two scenarios: Euclidean AdS2 evolution and Lorentzian dS2 evolution. It shows that naive holographic entropy calculations lead to islands and a strong subadditivity paradox, which are resolved by introducing bra-ket wormholes that connect the bra and ket sectors, yielding a consistent entanglement structure and a pure global state. In the AdS2 setup, bra-ket wormholes produce a thermofield-double-like interior and constrain long-interval entropies to saturate at 2S0, while removing paradoxes; in the compact case, the wormhole can dominate and maintains purity. Extending to dS2, several contours (identity, 2π, π) are explored, with the π contour and its bra-ket wormhole offering the most promising resolution, though issues remain—such as IR divergences and the need for a UV completion. Overall, bra-ket wormholes provide a compelling mechanism to reconcile gravitational entanglement structure with fundamental entropy inequalities and offer a fertile framework for connecting cosmology with holographic descriptions in two dimensions.

Abstract

We consider two dimensional CFT states that are produced by a gravitational path integral. As a first case, we consider a state produced by Euclidean $AdS_2$ evolution followed by flat space evolution. We use the fine grained entropy formula to explore the nature of the state. We find that the naive hyperbolic space geometry leads to a paradox. This is solved if we include a geometry that connects the bra with the ket, a bra-ket wormhole. The semiclassical Lorentzian interpretation leads to CFT state entangled with an expanding and collapsing Friedmann cosmology. As a second case, we consider a state produced by Lorentzian $dS_2$ evolution, again followed by flat space evolution. The most naive geometry also leads to a similar paradox. We explore several possible bra-ket wormholes. The most obvious one leads to a badly divergent temperature. The most promising one also leads to a divergent temperature but by making a projection onto low energy states we find that it has features that look similar to the previous Euclidean case. In particular, the maximum entropy of an interval in the future is set by the de Sitter entropy.

Figures (33)

• Figure 1: The two main cases we study. (a) We model an inflationary spacetime as follows. We consider a two dimensional de Sitter JT gravity theory coupled to a matter CFT$_2$. Inflation ends when the dilaton reaches a large value. We then transition to a flat space region, where we neglect the effects of gravity. This procedure produces a certain quantum state for the CFT$_2$ in the flat space region. (b) Another way to produce a state in a flat space CFT. We do Euclidean evolution in Euclidean AdS$_2$ JT gravity followed by Euclidean evolution in flat space. We impose transparent boundary conditions for the CFT$_2$ at the $AdS_2$ boundary.
• Figure 2: (a) $AdS_2$ with gravity plus flat space. (b) Holographic dual consisting of a flat space CFT interacting with some boundary degrees of freedom. (c) Euclidean rotation. We prepare a state in the CFT by doing Euclidean evolution in the gravity theory. (d) Boundary state interpretation.
• Figure 3: We consider a region $A$ in the state produced by the gravitational (green) plus the field theory (yellow) Euclidean evolution. (a) The naive result with no islands which just gives the field theory entropy of the interval in the vacuum. (b) The entropy when there are non-trivial quantum extremal surfaces. When $\ell$ is large the two intervals, represented by the dotted lines, are very far away. (c) The full computation involves two sides corresponding to the bra and the ket, here represented above and below the red interval. We can put the quantum extremal surface on either side indpendently for each of the two extremal surfaces.
• Figure 4: We display the quantum extremal surfaces involved in the computation of the various entropies. Though we are computing them at $\tau=0$ we included a bit of the (yellow) flat space region to emphasize that the subsystems are defined in the region with no gravity.
• Figure 5: (a) We consider a (green) wormhole geometry which connects two asymptotic boundaries, each of which is connected to the edges of a flat (yellow) strip. (b) In the dual description, we take the bra and the ket of the boundary state and compute the trace after doing some Euclidean evolution. In other words we compute $\langle B(\tau) | B(\tau) \rangle$. We can compute expectation values of operators by inserting them in the flat space region.