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The strong subadditivity of holographic entanglement entropy ; from boundary to bulk

Ali Izadi Rad

Published Jan 1, 2017• 0 citations
Publisher

Abstract

One decade ago, Ryu-Takayanagi explicitly introduced a formula that relates the entropy of a subregion in CFT system to a geometrical quantity which is called minimal surface in hyperbolic space. This formula extended to the idea of connection of gravy to quantum mechanics of gauge/gravity duality. This duality which can help us to learn a more interesting feature of each side from the other. Quantum systems obey from some constraints come from the quantum information theory. I would be interesting to find out what is the dual of this constraint in the gravitation system. Dual to the specific class of quantum theories which is called conformal field theories. One of the most significant constraint that QFTs should obey is the strong subadditivity of entanglement entropy. These constraints let the theories have bound on the energy spectrum from the below; Recently there has been the development that the combination of monotonicity of relative entropy and the strong subadditivity of entanglement entropy is equal to have a specific bound on the energy momentum tensor, called quantum null energy condition. In this thesis, we re-look to this argument by introducing the entanglement density and obtain a differential operator from the strong subadditivity and exploiting from the Markov property of the vacuum of CFT. In the next step, by using from the Ryu-Takayangi, we rewrite the strong subadditivity inequality regarding geometrical quantities. By using from the kinematic languages and intertwinement, we realize that the strong subadditivity at the boundary implies new bound on averaged energy condition which has some common feature with the quantum null energy condition statement.

DOI
10.14288/1.0355251

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One decade ago, Ryu-Takayanagi explicitly introduced a formula that relates the entropy of a subregion in CFT system to a geometrical quantity which is called minimal surface in hyperbolic space. This formula extended to the idea of connection of gravy to quantum mechanics of gauge/gravity duality. This duality which can help us to learn a more interesting feature of each side from the other. Quantum systems obey from some constraints come from the quantum information theory. I would be interesting to find out what is the dual of this constraint in the gravitation system. Dual to the specific class of quantum theories which is called conformal field theories. One of the most significant constraint that QFTs should obey is the strong subadditivity of entanglement entropy. These constraints let the theories have bound on the energy spectrum from the below; Recently there has been the development that the combination of monotonicity of relative entropy and the strong subadditivity of entanglement entropy is equal to have a specific bound on the energy momentum tensor, called quantum null energy condition. In this thesis, we re-look to this argument by introducing the entanglement density and obtain a differential operator from the strong subadditivity and exploiting from the Markov property of the vacuum of CFT. In the next step, by using from the Ryu-Takayangi, we rewrite the strong subadditivity inequality regarding geometrical quantities. By using from the kinematic languages and intertwinement, we realize that the strong subadditivity at the boundary implies new bound on averaged energy condition which has some common feature with the quantum null energy condition statement.
The holographic Ryu-Takayanagi formula for entanglement entropy connects the entanglement of a field theory to the geometry of a dual gravitational theory in a straightforward and universal way. The first part of this thesis applies this formula to study the entanglement entropy in strongly coupled noncommutative field theories. It is found that the ground state of these theories have substantial entanglement at the length scale of the noncommutativity. The entanglement entropy in a different perturbative regime is also computed, where in contrast it is found that noncommutative interactions do not induce long range entanglement in the ground state to leading order in perturbations theory. The second part of this thesis explores some general consequences of this holographic formula for the entanglement entropy. Identities involving entanglement entropies are related to nontrivial geometric constraints on gravitational duals. In particular, the strong subadditivity of entanglement entropy is used to show that dual three dimensional asymptotically anti-de Sitter gravitational states must obey an averaged null energy condition. Finally, this holographic formula allows us at least in principle to express the entanglement entropy of a region in a holographic field theory in terms of the one-point functions in that theory. This is explored in the context of a two dimensional conformal field theory where explicit calculations are possible. Our results in this case allow us to extend a recent proposal that the entanglement entropy of states near the vacuum of conformal theories can be understood by propagation in an auxiliary de Sitter space.
Abstract Using the Hubeny-Rangamani-Takayanagi (HRT) conjectured formula for entanglement entropy in the context of the AdS/CFT correspondence with time-dependent backgrounds, we investigate the relation between the bulk null energy condition (NEC) of the stress-energy tensor with the strong sub-additivity (SSA) property of entanglement entropy in the boundary theory. In a background that interpolates between an AdS to an AdS-Reissner-Nordstrom-type geometry, we find that generically there always exists a critical surface beyond which the violation of NEC would naively occur. However, the extremal area surfaces that determine the entanglement entropy for the boundary theory, can penetrate into this forbidden region only for certain choices for the mass and the charge functions in the background. This penetration is then perceived as the violation of SSA in the boundary theory. We also find that this happens only when the critical surface lies above the apparent horizon, but not otherwise. We conjecture that SSA, which is thus non- trivially related to NEC, also characterizes the entire time-evolution process along which the dual field theory may thermalize.
The Ryu-Takayanagi formula relates the entanglement entropy in a conformal field theory to the area of a minimal surface in its holographic dual. We show that this relation can be inverted for any state in the conformal field theory to compute the bulk stress-energy tensor near the boundary of the bulk spacetime, reconstructing the local data in the bulk from the entanglement on the boundary. We also show that positivity, monotonicity, and convexity of the relative entropy for small spherical domains between the reduced density matrices of any state and of the ground state of the conformal field theory are guaranteed by positivity conditions on the bulk matter energy density. As positivity and monotonicity of the relative entropy are general properties of quantum systems, this can be interpreted as a derivation of bulk energy conditions in any holographic system for which the Ryu-Takayanagi prescription applies. We discuss an information theoretical interpretation of the convexity in terms of the Fisher metric.
The Ryu-Takayanagi formula implies many general properties of entanglement entropies in holographic theories. We review the known properties, such as continuity, strong subadditivity, and monogamy of mutual information, and fill in gaps in some of the previously-published proofs. We also add a few new properties, including: properties of the map from boundary regions to bulk regions implied by the RT formula, such as monotonicity; conditions under which subadditivity-type inequalities are saturated; and an inequality concerning reflection-symmetric states. We attempt to draw lessons from these properties about the structure of the reduced density matrix in holographic theories.
In this Dissertation, different aspects of holographic entanglement entropy are explored. In quantum information theory, entanglement is a useful resource needed to perform quantum operations. Entanglement entropy quantifies this resource and even if computable in quantum field theories, it is quite hard to calculate explicitly. However, in the context of gauge/gravity duality, (holographic) entanglement entropy was proposed by Ryu and Takayanagi (RT) to be captured by the area of a minimal surface in Anti DeSitter (AdS) space, which is extremely simple to compute. We begin by using the holographic dictionary to derive the RT formula. This is done by first considering smooth bulk geometries which are dual to n copies of the boundary geometry glued in a particular way. The action of these geometries computes the Renyi entropies S_n for any integer n. We give a prescription to analytically continue the action to non-integer n and argue that, as n goes to 1,S_1, the entanglement entropy, is given by the area of a minimal surface, reproducing the RT conjecture. That is, we reduced the RT proposal to the equality between bulk and boundary partition functions, a standard entry in the dictionary. We then generalize the previous formalism to account for corrections due to bulk quantum fields (1/N corrections in the boundary). This allows us to derive a new formula: boundary entanglement entropy is given by the area of the minimal surface plus bulk entanglement entropy. This extends RT beyond the planar limit and we present several predictions of the proposal. Our exploration of holographic entanglement entropy continues by considering different states: by comparing the quantum corrected entropy for nearby states we obtain the modular hamiltonian operator in bulk perturbation theory. From this expression of the modular hamiltonian, we derive that relative entropy is bulk relative entropy and that the modular flow is the bulk modular flow.
Significant work has gone into determining the minimal set of entropy inequalities that determine the holographic entropy cone. Holographic systems with three or more parties have been shown to obey additional inequalities that generic quantum systems do not. We consider a two dimensional conformal field theory that is a single boundary of a holographic system and find four additional non-linear inequalities which are derived from strong subadditivity and the formula for the entanglement entropy of a region on the conformal field theory. We also present an equality obtained by application of a hyperbolic extension of Ptolemy's theorem to a two dimensional conformal field theory.