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Mark Van Raamsdonk
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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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These lectures review recent developments in our understanding of the emergence of local bulk physics in AdS/CFT. The primary topics are sufficient conditions for a conformal field theory to have a semiclassical dual, bulk reconstruction, the quantum error correction interpretation of the correspondence, tensor network models of holography, and the quantum Ryu-Takayanagi formula.
We focus on two types of coherent states, the coherent states of multi graviton states and the coherent states of giant graviton states, in the context of gauge/gravity correspondence. We conveniently use a phase shift operator and its actions on the superpositions of these coherent states. We find $N$-state Schrodinger cat states which approach the one-row Young tableau states, with fidelity between them asymptotically reaches 1 at large $N$. The quantum Fisher information of these states is proportional to the variance of the excitation energy of the underlying states, and characterizes the localizability of the states in the angular direction in the phase space. We analyze the correlation and entanglement between gravitational degrees of freedom using different regions of the phase space plane in bubbling AdS. The correlation between two entangled rings in the phase space plane is related to the area of the annulus between the two rings. We also analyze two types of noisy coherent states, which can be viewed as interpolated states that interpolate between a pure coherent state in the noiseless limit and a maximally mixed state in the large noise limit.
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.
The aim of these Lectures is to provide a brief overview of the subject of asymptotic symmetries of gauge and gravity theories in asymptotically flat spacetimes as background material for celestial holography.
In this paper, we demonstrate the emergence of nonlinear gravitational equations directly from the physics of a broad class of conformal field theories. We consider CFT excited states defined by adding sources for scalar primary or stress tensor operators to the Euclidean path integral defining the vacuum state. For these states, we show that up to second order in the sources, the entanglement entropy for all ball-shaped regions can always be represented geometrically (via the Ryu-Takayanagi formula) by an asymptotically AdS geometry. We show that such a geometry necessarily satisfies Einsteins equations perturbatively up to second order, with a stress energy tensor arising from matter fields associated with the sourced primary operators. We make no assumptions about AdS/CFT duality, so our work serves as both a consistency check for the AdS/CFT correspondence and a direct demonstration that spacetime and gravitational physics can emerge from the description of entanglement in conformal field theories.
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