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Remarks on the entanglement entropy for disconnected regions

Horacio Casini, Marina Huerta

Published Nov 26, 2012• 121 citations
OpenAlex

Abstract

Few facts are known about the entanglement entropy for disconnected regions in quantum field theory. We study here the property of extensivity of the mutual information, which holds for free massless fermions in two dimensions. We uncover the structure of the entropy function in the extensive case, and find an interesting connection with the renormalization group irreversibility. The solution is a function on space-time regions which complies with all the known requirements a relativistic entropy function has to satisfy. We show that the holographic ansatz of Ryu and Takayanagi, the free scalar and Dirac fields in dimensions greater than two, and the massive free fields in two dimensions all fail to be exactly extensive, disproving recent conjectures. 1

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Few facts are known about the entanglement entropy for disconnected regions in quantum field theory. We study here the property of extensivity of the mutual information, which holds for free massless fermions in two dimensions. We uncover the structure of the entropy function in the extensive case, and find an interesting connection with the renormalization group irreversibility. The solution is a function on space-time regions which complies with all the known requirements a relativistic entropy function has to satisfy. We show that the holographic ansatz of Ryu and Takayanagi, the free scalar and Dirac fields in dimensions greater than two, and the massive free fields in two dimensions all fail to be exactly extensive, disproving recent conjectures. 1
We study the entanglement entropy in a relativistic quantum field theory for regions which are not included in a single spatial hyperplane. This geometric configuration cannot be treated with the Euclidean time method and the replica trick. Instead, we use a real time method to calculate the entropy for a massive free Dirac field in two dimensions in some approximations. We find some specifically relativistic features of the entropy. First, there is a large enhancement of entanglement due to boosts. As a result, the mutual information between relatively boosted regions does not vanish in the limit of zero volume and large relative boost. We also find extensivity of the information in a deeply Lorentzian regime with large violations of the triangle inequalities for the distances. This last effect is relevant to an interpretation of the amount of entropy enclosed in the Hawking radiation emitted by a black hole.
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.
Entanglement entropy is a proposal to quantify quantum entanglement of two disjoint regions in a pure system. It is a relatively new topic and is developing rapidly. The current main motives to study it are its applications to studying quantum gravity, thermalization and phase transitions in condensed matter systems. Mutual information is a related quantity and it can be used to measure the amount of information two disjoint regions share. The purpose of this thesis is to give an introduction to the general results of the field without considering specific systems. The two prominent approaches used are two-dimensional conformal field theory and the holographic entanglement entropy conjecture. The first approach is to calculate entanglement entropy using conformal field theory, the results of which are known to be exact although technically more difficult to derive and only available for 1+1 dimensional systems. Both static and dynamic systems will be discussed. The results are reproduced and generalized to higher dimensions using holography. As a more recent topic, the holographic approach is used to rederive the entanglement entropy of systems with a Fermi surface using AdS/Vaidya metric with Lifshitz scaling and hyperscaling violation. The final chapter of the thesis discusses mutual information in various static and dynamic cases considered in the previous chapters. For the first time, the evolution of mutual information is calculated in AdS/Vaidya metric with Lifshitz scaling and hyperscaling violation in the critical theta=d-1 case.
First, we study the entanglement entropy S AB of a d + 1 dimensional massless free scalar field on two disjoint compact spatial regions A and B whose sizes are R 1 and R 2 , respectively, and the distance between them is r.The state of the massless free scalar field is the vacuum state.We consider the mutual information S A;B ≡ S A + S B -S AB where S A and S B are the entanglement entropy on the inside region of A and B, respectively.We develop the computational method based on that of Bombelli et al and obtain the result thatWhen A and B are spheres in d = 2 and 3, we obtain explicit resultsBy using the method used in the Minkowski spacetime case with some modifications, we study the entanglement entropy, S C , of the massless free scalar field on the outside region C of two black holes A and B whose radii are R 1 and R 2 and how it depends on the distance, r( R 1 , R 2 ) , between two black holes.If we can consider the entanglement entropy as thermodynamic entropy, we can see the entropic force (we call this entanglement entropic force) acting on the two black holes from the r dependence of S C .We consider the property of the entropic force.
These notes, based on lectures given at various schools over the last few years, aim to provide an introduction to entanglement entropies in quantum field theories, including holographic ones. We explore basic properties and simple examples of entanglement entropies, mostly in two dimensions, with an emphasis on physical rather than formal aspects of the subject. In the holographic case, the focus is on how the Ryu-Takayanagi formula geometrically realizes general features of field-theory entanglement, while revealing special properties of holographic theories. In order to make the notes somewhat self-contained for readers whose background is in high-energy theory, a brief introduction to the relevant aspects of quantum information theory is included.
Ryu and Takayanagi conjectured a formula for the entanglement (von Neumann) entropy of an arbitrary spatial region in an arbitrary holographic field theory. The von Neumann entropy is a special case of a more general class of entropies called R\'enyi entropies. Using Euclidean gravity, Fursaev computed the entanglement R\'enyi entropies (EREs) of an arbitrary spatial region in an arbitrary holographic field theory, and thereby derived the RT formula. We point out, however, that his EREs are incorrect, since his putative saddle points do not in fact solve the Einstein equation. We remedy this situation in the case of two-dimensional conformal field theories (CFTs), considering regions consisting of one or two intervals. For a single interval, the EREs are known for a general CFT; we reproduce them using gravity. For two intervals, the RT formula predicts a phase transition in the entanglement entropy as a function of their separation, and that the mutual information between the intervals vanishes for separations larger than the phase transition point. By computing EREs using gravity and CFT techniques, we find evidence supporting both predictions. We also find evidence that large NNN symmetric product theories have the same EREs as holographic ones.