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 ∪ 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.
Highlights
AdS2 geometryWe will analyze pure AdS JT gravity and consider the back reactions to the dilaton and the geometry from the conformal matter and the initial pure state
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
In order to find the entanglement island, we must look for the replica wormhole geometry, which gives the dominant contribution to the replica partition function TrρnA for a sufficiently large subregion A
Summary
We consider the simplest example; Euclidean JT gravity [102]. We set Λ = −1. The boundary action for externally fixed h and φ on P is. K and ∂nφ − T φ are fixed on-shell. The tensions T and T , and the action (2.6) can be derived via dimensional reduction from higher dimensional EOW brane model [104], see [9, 10, 105, 106]. When the two dimensional model is derived by such dimensional reduction, T is related to T as T = φ0T. This implies that in order to have finite dilaton φ, we need to impose K = T = O(φ−0 1)
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