Abstract
We consider CFT states defined by adding nonlocal multi-trace sources to the Euclidean path integral defining the vacuum state. For holographic theories, we argue that these states correspond to states in the gravitational theory with a good semiclassical description but with a more general structure of bulk entanglement than states defined from single-trace sources. We show that at leading order in large N , the entanglement entropies for any such state are precisely the same as those of another state defined by appropriate single-trace effective sources; thus, if the leading order entanglement entropies are geometrical for the single-trace states of a CFT, they are geometrical for all the multi-trace states as well. Next, we consider the perturbative calculation of 1/N corrections to the CFT entanglement entropies, demonstrating that these show qualitatively different features, including non-analyticity in the sources and/or divergences in the naive perturbative expansion. These features are consistent with the expectation that the 1/N corrections include contributions from bulk entanglement on the gravity side. Finally, we investigate the dynamical constraints on the bulk geometry and the quantum state of the bulk fields which must be satisfied so that the entropies can be reproduced via the quantum-corrected Ryu-Takayanagi formula.
Highlights
In recent years, it has become clear that the emergence of gravitational physics from certain non-gravitational quantum systems with large N,1 as suggested by the AdS/CFT correspondence [1,2,3], can be made transparent by considering the structure of entanglement in the non-gravitational system
We show that at leading order in large N, the entanglement entropies for any such state are precisely the same as those of another state defined by appropriate single-trace effective sources; if the leading order entanglement entropies are geometrical for the single-trace states of a CFT, they are geometrical for all the multitrace states as well
In [14], these results were extended to second-order perturbations: for a class of CFT states produced by adding local sources to the Euclidean path integral that describes the vacuum state, the ball entanglement entropies up to second order in the sources defining the state can be captured geometrically by a second-order perturbation to AdS
Summary
It has become clear that the emergence of gravitational physics from certain non-gravitational quantum systems with large N ,1 as suggested by the AdS/CFT correspondence [1,2,3], can be made transparent by considering the structure of entanglement in the non-gravitational system (see, e.g., [4,5,6,7,8]). For states with a dual gravitational description, entanglement entropies for spatial subsystems (considered at leading order in N ) are related to the areas of extremal surfaces in the corresponding spacetime [5, 9]. For arbitrary first-order perturbations to the vacuum state, the ball entanglement entropies can always be captured by extremal surface areas in some first-order perturbation of this AdS spacetime. The Entanglement First Law [10] implies that the perturbed spacetime geometry must satisfy Einstein’s equations linearized about AdS [11, 12].2. The structure of CFT entanglement (in particular, its relation to the one-point functions of local operators) implies that these perturbations must satisfy local gravitational equations, which include nonlinear contributions.
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