Island for Gravitationally Prepared State and Pseudo Entanglement Wedge

TL;DR

This paper explores a generalized spacetime with finite initial boundary, analyzing entanglement entropy and introducing the concept of pseudo entanglement wedges, which extend entanglement wedge reconstruction in holography.

Contribution

It introduces the pseudo entanglement wedge concept, generalizes entanglement entropy bounds with islands, and studies the reconstruction of bulk matter in spacetimes with finite initial boundaries.

Findings

Entanglement entropy is bounded by the boundary area of the island.

A necessary condition for initial states to satisfy strong sub-additivity.

Reconstruction of bulk matter transition matrix is exponentially hard, extending Python's lunch conjecture.

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.