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arXiv:0704.3719 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Holographic proof of the strong subadditivity of entanglement entropy

Matthew Headrick, Tadashi Takayanagi

Published Nov 21, 2007• 298 citations
arXivHTMLPDF

Abstract

When a quantum system is divided into subsystems, their entanglement entropies are subject to an inequality known as strong subadditivity. For a field theory this inequality can be stated as follows: given any two regions of space $A$ and $B$, $S(A)+S(B)\ensuremath{\ge}S(A\ensuremath{\cup}B)+S(A\ensuremath{\cap}B)$. Recently, a method has been found for computing entanglement entropies in any field theory for which there is a holographically dual gravity theory. We give a simple geometrical proof of strong subadditivity employing this holographic prescription.

DOI
10.1103/physrevd.76.106013

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When a quantum system is divided into subsystems, their entanglement entropies are subject to an inequality known as strong subadditivity. For a field theory this inequality can be stated as follows: given any two regions of space AAA and BBB, S(A)+S(B)\ensuremath≥S(A\ensuremath∪B)+S(A\ensuremath∩B)S(A)+S(B)\ensuremath{\ge}S(A\ensuremath{\cup}B)+S(A\ensuremath{\cap}B)S(A)+S(B)\ensuremath≥S(A\ensuremath∪B)+S(A\ensuremath∩B). Recently, a method has been found for computing entanglement entropies in any field theory for which there is a holographically dual gravity theory. We give a simple geometrical proof of strong subadditivity employing this holographic prescription.
Abstract The vacuum entanglement entropy in quantum field theory provides nonperturbative information about renormalization group flows. Most studies so far have focused on the universal terms, related to the Weyl anomaly in even space-time dimensions, and the sphere free energy F in odd dimensions. In this work we study the entanglement entropy on a sphere of radius R in a large radius limit, for field theories with gravity duals. At large radius the entropy admits a geometric expansion in powers of R; the leading term is the well-known area term, and we also consider the subleading contributions. These terms can be physical, they contain information about the full renormalization group flow, and they reproduce known monotonicity theorems in particular cases. We set up an efficient method for calculating them using the Hamilton-Jacobi equation for the holographic entanglement entropy. We first reproduce the known result for the area term, the coefficient multiplying Rd−2 in the entanglement entropy. We then obtain the holographic result for the Rd−4 term and establish its irreversibility. Finally, we derive the Rd−6 coefficient for holographic theories, and also establish its irreversibility. This result goes beyond what has been proved in quantum field theory based on strong subadditivity, and hints towards new methods for analyzing the monotonicity of the renormalization group in space-time dimensions bigger than four.
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.
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.
Strong subadditivity (SSA) on a common light cone gives the Casini-Huerta entropic proof of the three-dimensional FFF-theorem. We ask whether holographic entropy inequalities beyond SSA similarly constrain renormalization group flows for which every intermediate theory admits a semiclassical holographic description. For small, disjoint deformations of a common light-cone region, we show that the second-order response of a broad class of balanced holographic inequalities depends only on pairwise correlations already controlled by SSA. A six-party example shows that the full finite inequality nevertheless contains genuinely multipartite information, so its disappearance is a limitation of the second-order expansion rather than of the inequality itself. Two natural finite constructions do not recover the missing information. We nevertheless find two ways in which information beyond SSA survives. A five-party inequality bounds the rate at which a conditional correlation grows as one region is enlarged. Separately, a continuum limit of the odd-cyclic inequalities gives a constraint on the angular shape dependence of entanglement entropy, and Lorentz symmetry relates this constraint to radial evolution. Thus holographic entropy inequalities beyond SSA do constrain entanglement along RG flows, although we do not obtain a second universal analogue of the FFF-function.
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.
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.