Entanglement entropy for non-coplanar regions in quantum field theory

Abstract

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. *

Holographic quantum circuits from splitting/joining local quenches

“…Then, by summing all bipartite entanglement, using the entanglement density (A.1), S A is estimated as follows: This indeed agrees with the known expression of entanglement entropy in the massless Dirac fermion CFT [7]. A closely related property is that the multi partite mutual information is vanishing [76].…”

Comments on Jacobson’s “entanglement equilibrium and the Einstein equation”

“…In fact, in section 2.3.1, we confirmed the appearance of this extra logarithmic dependence beyond the framework of holography. There, we found that that the same term appears for a free Dirac fermion in two dimensions, for which the entanglement Hamiltonian is explicitly known [41,42]. When the theory is perturbed by a small mass, i.e., mR ≪ 1, it is quite remarkable that the same logarithmic term appears in eq.…”

“…It is important to understand the appearance of this term is special to such a holographic framework or if similar terms arise with general CFT's. In the case of the free massive Dirac fermion in d = 2, the modular Hamiltonian is known exactly and so can be computed perturbatively for small mass [41,42]. In this case, the mass operator has dimension ∆ = d − 1 = 1 = d/2 and so it is possible to check whether the logarithmic behaviour is also present in that context.…”

Entanglement entropy of excited states in conformal perturbation theory and the Einstein equation

“…Furthermore, for regular states, δ T 0 00 is UV finite, and hence the answer may be written without reference to the renormalization scale as in (4.48), although it explicitly depends on the IR cutoff. In some cases, such as free field theories, the appropriate IR cutoff may be calculated exactly [25,48,49]. Reexpressing the answer in terms of O g instead of the IR cutoff, as in equation (1.2), re-introduces the renormalization scale µ, since the vev requires renormalization and hence is µ-dependent.…”