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arXiv:1312.6717 · Journal of High Energy Physics

General properties of holographic entanglement entropy

Matthew Headrick

Published Mar 1, 2014• 85 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/jhep03(2014)085

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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.
We identify a special information-theoretic property of quantum field theories with holographic duals: the mutual informations among arbitrary disjoint spatial regions AAA, BBB, CCC obey the inequality $I(A\ensuremath{\mathbin:}B\ensuremath{\cup}C)\ensuremath{\ge}I(A\ensuremath{\mathbin:}B)+I(A\ensuremath{\mathbin:}C)$, provided entanglement entropies are given by the Ryu-Takayanagi formula. Inequalities of this type are known as monogamy relations and are characteristic of measures of quantum entanglement. This suggests that correlations in holographic theories arise primarily from entanglement rather than classical correlations. We also show that the Ryu-Takayanagi formula is consistent with all known general inequalities obeyed by the entanglement entropy, including an infinite set recently discovered by Cadney et al.; this constitutes strong evidence in favor of its validity.
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.
Entanglement entropies computed using the holographic Ryu-Takayanagi formula are known to obey an infinite set of linear inequalities, which define the so-called RT entropy cone. The general structure of this cone, or equivalently the set of all valid inequalities, is unknown. It is also unknown whether those same inequalities are also obeyed by entropies computed using the covariant Hubeny-Rangamani-Takayanagi formula, although significant evidence has accumulated that they are. Using Markov states, we develop a test of this conjecture in a heretofore unexplored regime. The test reduces to checking that a given inequality obeys a certain majorization property, which is easy to evaluate. We find that the RT inequalities pass this test and, surprisingly, only such inequalities do so. Our results not only provide strong new evidence that the HRT and RT cones coincide, but also offer a completely new characterization of that cone.
As a consequence of the Ryu-Takayanagi formula, static semiclassical states in AdS quantum gravity share a set of special properties expressible as linear inequalities on their entanglement entropies. How general are these properties? Do they still hold when we move away from the regime of validity of RT, i.e. static states in classical Einstein gravity? Do they perhaps hold even for some class of non-holographic states? We will motivate these questions and review what's known about the answers.
We explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in AdS3_33​/CFT2_{2}2​. With the guidance from the boundary and bulk modular flows we find a natural slicing of the entanglement wedge with the modular planes, which are co-dimension one bulk surfaces tangent to the modular flow everywhere. This gives an one-to-one correspondence between the points on the boundary interval A\mathcal{A}A and the points on the Ryu-Takayanagi (RT) surface EA\mathcal{E}_{\mathcal{A}}EA​. In the same sense an arbitrary subinterval A2\mathcal{A}_2A2​ of A\mathcal{A}A will correspond to a subinterval E2\mathcal{E}_2E2​ of EA\mathcal{E}_{\mathcal{A}}EA​. We propose that the length of E2\mathcal{E}_2E2​ captures the contribution from A2\mathcal{A}_2A2​ to the entanglement entropy SAS_{\mathcal{A}}SA​, and compare it with the mutual information between A2\mathcal{A}_2A2​ and the complement Ac\mathcal{A}^cAc.
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.