Island for Gravitationally Prepared State and Pseudo Entanglement Wedge

TL;DR

This work analyzes entanglement in cosmological spacetimes prepared by a finite initial boundary within Euclidean JT gravity, showing that large-subregion entropy is bounded by an island or the initial boundary, formalized via a pseudo-entropy island formula. It develops a coherent framework combining an EOW-brane–based initial boundary, a Schwinger-Keldysh replica trick, and pseudo entropy to define a pseudo entanglement wedge and island. The paper introduces an AdS/BCFT model with corner terms (Hayward term) and demonstrates conditions under which the spacetime with finite initial boundary dominates over no-boundary or bra-ket wormholes, while ensuring consistency with strong sub-additivity. It also extends entanglement-wedge reconstruction to non-time-symmetric settings through bulk transition matrices and proposes a pseudo Python’s Lunch bound on the recoverability of island information, highlighting exponential hardness. The results illuminate how gravitational preparation shapes information bounds and reconstruction in cosmological holography and suggest directions for Lorentzian extensions and causality analyses.

Abstract

We consider spacetime initiated by a finite-sized initial boundary as a generalization of the Hartle-Hawking no-boundary state. We study entanglement entropy of matter state prepared by such spacetime. We find that the entanglement entropy for large subregion is given either by the initial state entanglement or the entanglement island, preventing the entropy to grow arbitrarily large. Consequently, the entanglement entropy is always bounded from above by the boundary area of the island, leading to an entropy bound in terms of the island. The island $I$ is located in the analytically continued spacetime, either at the bra or the ket part of the spacetime in Schwinger-Keldysh formalism. The entanglement entropy is given by an average of $complex$ pseudo generalized entropy for each entanglement island. We find a necessary condition of the initial state to be consistent with the strong sub-additivity, which requires that any probe degrees of freedom are thermally entangled with the rest of the system. We then find a large parameter region where the spacetime with finite-sized initial boundary, which does not have the factorization puzzle at leading order, dominates over the Hartle-Hawking no-boundary state or the bra-ket wormhole. Due to the absence of a moment of time reflection symmetry, the island in our setup is a generalization of the entanglement wedge, called pseudo entanglement wedge. In pseudo entanglement wedge reconstruction, we consider reconstructing the bulk matter transition matrix on $A\cup I$, from a fine-grained state on $A$. The bulk transition matrix is given by a thermofield double state with a projection by the initial state. We also provide an AdS/BCFT model by considering EOW branes with corners. We also find the exponential hardness of such reconstruction task using a generalization of Python's lunch conjecture to pseudo generalized entropy.

Figures (11)

• Figure 1: The spacetime we consider is initiated by an End-of-the-world brane. We consider the matter state prepared by such spacetime, and study its entanglement entropy. The entanglement entropy has three phases. ($a$) Naive thermal entropy phase, computed via effective matter theory. ($b$) The island phase, the entropy is an average of pseudo entropy. ($c$) The boundary phase, the entropy is given by the boundary entropy of the initial state.
• Figure 2: The replica geometry in Schwinger-Keldysh formalism, with a replica wormhole in the ket part of the bulk. This geometry computes the pseudo entanglement entropy $S^{\text{ket}}_A$. The average of two pseudo entanglement entropy gives the desired entanglement entropy $S^{\text{Island}}_A(I)=(S^{\text{ket}}_A+S^{\text{bra}}_A)/2$, assuming there are no other saddles.
• Figure 3: ($a$) The geometry for the bulk transition matrix $\rho_{\text{ket}}^{\text{Bulk}}(A\cup I)$. Its complex conjugate gives the bra counter part. The entanglement island is present only at the ket part, so the time reflection symmetry is absent. ($b$) The bulk transition matrix on $A\cup I$ can be constructed in terms of thermofield double state and by projection onto the initial state.
• Figure 4: Double wormholes connecting bra-and-bra and ket-and-ket. Taking $\mathbb{Z}_2$ orbifold of this two wormhole geometry gives the Schwinger-Keldysh geometry with an initial boundary state.
• Figure 5: Three phases of pseudo RT surface $\gamma$ for a boundary subregion. ($a$) $\gamma$ does not intersect with any of $\Sigma_1$, $\Sigma_2$ and $\Gamma$. ($b$) $\gamma$ ends on EOW branes $\Sigma_1$ and $\Sigma_2$. When the set up is time reflection symmetric, there are multiple pseudo RT surfaces which are related via time reflection. ($c$) $\gamma$ ends on the corner $\Gamma$.