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Register a new userEntanglement of Initial State and Pseudo Entanglement Wedge
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Masamichi Miyaji
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We consider spacetime initiated by a finite-sized boundary on which a pure initial matter state is set as a natural generalization of the Hartle-Hawking no-boundary state. We study entanglement entropy of the gravitationally prepared matter state at the final time slice. We find that the entropy of the initial state or the entanglement island gives the entropy for large subregions on the final time slice. Consequently, we find the entanglement entropy is bounded from above by the boundary area of the island, leading to an entropy bound in terms of the island formula. The island $I$ appears in the analytically continued spacetime, either at the bra or the ket part of the spacetime in Schwinger-Keldysh formalism, and the entropy is given by an average of pseudo entropy of each entanglement island. We find a necessary condition of the initial state to be consistent with the strong sub-additivity. The condition requires that any probe degrees of freedom are thermally entangled with the rest of the system. We then study which initial condition leads to our finite-sized initial boundary or the Hartle-Hawking no-boundary state. Due to the absence of a moment of time reflection symmetry, the island in our setup requires a generalization of the entanglement wedge, which we call {it{pseudo entanglement wedge}}. In pseudo entanglement wedge reconstruction, we consider reconstructing the bulk matter transition matrix on $Acup I$, 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 provide an AdS/BCFT model, which provides a double holography model of our setup by considering EOW branes with corners. We also find the exponential hardness of such reconstruction task using a generalization of Pythons lunch conjecture to pseudo generalized entropy.
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In the holographic correspondence, subregion duality posits that knowledge of the mixed state of a finite spacelike region of the boundary theory allows full reconstruction of a specific region of the bulk, known as the entanglement wedge. This statement has been proven for local bulk operators. In this paper, specializing first for simplicity to a Rindler wedge of AdS$_3$, we find that generic curves within the wedge are in fact not fully reconstructible with entanglement entropies in the corresponding boundary region, even after using the most general variant of hole-ography, which was recently shown to suffice for reconstruction of arbitrary spacelike curves in the Poincare patch. This limitation is an analog of the familiar phenomenon of entanglement shadows, which we call entanglement shade. We overcome it by showing that the information about the nonreconstructible curve segments is encoded in a slight generalization of the concept of entanglement of purification, whose holographic dual has been discussed very recently. We introduce the notion of differential purification, and demonstrate that, in combination with differential entropy, it enables the complete reconstruction of all spacelike curves within an arbitrary entanglement wedge in any 3-dimensional bulk geometry.
We study the entanglement wedge cross-section (EWCS) in holographic Aether gravity theory, a gravity theory with Lorentz symmetry breaking meanwhile keeping the general covariance intact. We find that only a limited parameter space is allowed to obtain a black brane with positive Hawking temperature. Subject to these allowed parameter regions, we find that the EWCS could exhibit non-monotonic behaviors with system parameters. Meanwhile, the holographic entanglement entropy (HEE), and the corresponding mutual information (MI), can only exhibit monotonic behaviors. These phenomena suggest that the EWCS could capture much more rich content of the entanglement than that of the HEE and the MI. The role of the Lorentz violation in determining the behaviors of quantum information-related quantities is also analyzed.
We study the mixed state entanglement properties in two holographic axion models by examining the behavior of the entanglement wedge minimum cross section (EWCS), and comparing it with the holographic entanglement entropy (HEE) and mutual information (MI). We find that the behavior of HEE, MI and EWCS with Hawking temperature is monotonic, while the behavior with the axion parameter $k$ is more rich, which depends on the size of the configuration and the values of the other two parameters. Interestingly, the EWCS monotonically increases with the coupling constant $kappa$ between the axion field and the Maxwell field, while HEE and MI can be non-monotonic. It suggests that the EWCS, as a mixed state entanglement measure, captures distinct degrees of freedom from the HEE and MI indeed. We also provide analytical understandings for most of the numerical results.
We argue that holographic CFT states require a large amount of tripartite entanglement, in contrast to the conjecture that their entanglement is mostly bipartite. Our evidence is that this mostly-bipartite conjecture is in sharp conflict with two well-supported conjectures about the entanglement wedge cross section surface $E_W$. If $E_W$ is related to either the CFTs reflected entropy or its entanglement of purification, then those quantities can differ from the mutual information at $mathcal{O}(frac{1}{G_N})$. We prove that this implies holographic CFT states must have $mathcal{O}(frac{1}{G_N})$ amounts of tripartite entanglement. This proof involves a new Fannes-type inequality for the reflected entropy, which itself has many interesting applications.
The AdS/CFT correspondence provides quantum theories of gravity in which spacetime and gravitational physics emerge from ordinary non-gravitational quantum systems with many degrees of freedom. Recent work in this context has uncovered fascinating connections between quantum information theory and quantum gravity, suggesting that spacetime geometry is directly related to the entanglement structure of the underlying quantum mechanical degrees of freedom and that aspects of spacetime dynamics (gravitation) can be understood from basic quantum information theoretic constraints. In these notes, we provide an elementary introduction to these developments, suitable for readers with some background in general relativity and quantum field theory. The notes are based on lectures given at the CERN Spring School 2014, the Jerusalem Winter School 2014, the TASI Summer School 2015, and the Trieste Spring School 2015.
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