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)