Relative entropy and the RG flow

Journal of High Energy Physics, Mar 2017

Abstract We consider the relative entropy between vacuum states of two different theories: a conformal field theory (CFT), and the CFT perturbed by a relevant operator. By restricting both states to the null Cauchy surface in the causal domain of a sphere, we make the relative entropy equal to the difference of entanglement entropies. As a result, this difference has the positivity and monotonicity properties of relative entropy. From this it follows a simple alternative proof of the c-theorem in d = 2 space-time dimensions and, for d > 2, the proof that the coefficient of the area term in the entanglement entropy decreases along the renormalization group (RG) flow between fixed points. We comment on the regimes of convergence of relative entropy, depending on the space-time dimensions and the conformal dimension Δ of the perturbation that triggers the RG flow.

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Relative entropy and the RG flow

Received: February Relative entropy and the RG ow Horacio Casini 0 1 Eduardo Teste 0 1 Gonzalo Torroba 0 1 Centro Atomico Bariloche 0 1 CONICET 0 1 Open Access 0 1 c The Authors. 0 1 0 xed points. We comment 1 S.C. de Bariloche , R o Negro, R8402AGP , Argentina We consider the relative entropy between vacuum states of two di erent theories: a conformal eld theory (CFT), and the CFT perturbed by a relevant operator. By restricting both states to the null Cauchy surface in the causal domain of a sphere, we make the relative entropy equal to the di erence of entanglement entropies. As a result, this di erence has the positivity and monotonicity properties of relative entropy. From this it follows a simple alternative proof of the c-theorem in d = 2 space-time dimensions and, for d > 2, the proof that the coe cient of the area term in the entanglement entropy decreases along the renormalization group (RG) ow between on the regimes of convergence of relative entropy, depending on the space-time dimensions and the conformal dimension of the perturbation that triggers the RG ow. ow; Renormalization Group; Conformal Field Theory 1 Introduction Relative entropy for states of di erent theories Reduction to a spatial region of two states of di erent theories Conformal interaction picture Modular Hamiltonian The null limit Entanglement entropy and regimes of relative entropy Consequences for the entanglement entropy A simple proof of the c-theorem Monotonicity of the area term in entanglement entropy eld examples A.1 Massless and massive scalar elds A.2 Boundary term in the modular Hamiltonian Introduction The renormalization group (RG) ow describes how physics changes with scale in a quan eld theory (QFT). In recent years, interesting connections of these ows with quantum information theory (QIT) have been discovered. A universal term in the vacuum entanglement entropy (EE) was shown to decrease monotonically along the RG for space-time the QFT, the key property of these proofs is strong subadditivity of entanglement entropy. Holographically, the monotonicity of the RG ow is related to the null energy condition in the bulk [5, 7]. More generally, the ne-grained RG ow in terms of tensor networks [8] has been proposed as a description of the spatial structure of the holographic gravity dual [9]. A natural information theory tool to study changes between states is the relative entropy. This meassures distinguishability between di erent states in a precise operational way [10]. In the context of the renormalization group ows a natural idea is to use relative entropy to quantify how a theory (or its vacuum state) gets modi ed as we change the Relative entropy has also started to play important roles in black hole physics, holography and quantum eld theory; see e.g. [13{18]. 1Previous steps in this direction include [11, 12], who studied the classical relative entropy between the probability distributions de ned by the Euclidean path integrals as a measure of distinguishability between theories. A change in the Lagrangian that induces the RG ow produces a change in the state associated to the path integral probability distribution. In this work we consider quantum relative entropies in real time, between vacuum states of two theories reduced to certain regions, and look at the consequences of positivity and monotonicity of relative entropy. We follow the steps of the recent work [19], where Evidently, not every pair of vacuum states of two di erent theories can be compared through the relative entropy. Di erent theories, i.e. containing one and two free scalar elds respectively, usually live in di erent Hilbert spaces, and there is no natural meaning in taking a relative entropy in this case. In order to compute a relative entropy, we need that (at least in presence of a physical UV cuto such as a lattice) the microscopic constituents of the two models be the same. For this reason, we will study theories with the same UV xed point, where this can in principle be achieved. More precisely, we will x as a reference state the UV conformal xed point itself, and study the relative entropy with another state arising from the CFT by perturbing it with a relevant operator. We will argue that relative entropy gives a useful notion of statistical distance between these theories, and is well-suited for capturing global properties of RG Relative entropy is notoriously e cient in distinguishing states. It essentially takes into account all ne grained information about the states. In our setup this is re ected in the possible presence of divergences. In order to get de nite results for RG ows, we need to avoid these divergences and prevent the relative entropy from distinguishing the states Divergences may be of UV origin, due to the fact that even if the two theories we consider approach each other at short distances, the correlators of the deformed theory do not converge to the ones of the CFT fast enough to make (...truncated)


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Horacio Casini, Eduardo Testé, Gonzalo Torroba. Relative entropy and the RG flow, Journal of High Energy Physics, 2017, pp. 89, Volume 2017, Issue 3, DOI: 10.1007/JHEP03(2017)089