Petz-Rényi relative entropy in QFT from modular theory

TL;DR

The paper defines and analyzes the Petz–Rényi relative entropy $\\mathcal{S}_{\\alpha}(\\Psi\\Vert\\Phi)$ for states of a von Neumann algebra using modular theory, proving well-definedness for $\\alpha\\in[0,1)$ and that $\\lim_{\\alpha\to1^-}\\mathcal{S}_{\\alpha}(\\Psi\\Vert\\Phi)=\\mathcal{S}(\\Psi\\Vert\\Phi)$. For unitary excitations of a common reference state, the entropy depends only on the reference state's modular data and can be computed via analytic continuation of the modular flow. In free bosonic QFTs, the Petz–Rényi entropy depends on the symmetric part of the two-point function $\\omega_2$ in addition to the symplectic form, making it genuinely quantum and expressible in the standard subspace framework. The authors provide explicit computations for a free scalar in the Minkowski wedge and for a free chiral current in a thermal state, including temperature dependence and asymptotics, and establish monotonicity and bounds relative to the Araki–Uhlmann entropy.

Abstract

We consider the generalization of the Araki-Uhlmann formula for relative entropy to Petz-Rényi relative entropy. We compute this entropy for a free scalar field in the Minkowski wedge between the vacuum and a coherent state, as well as for the free chiral current in a thermal state. In contrast to the relative entropy which in these cases only depends on the sympletic form and thus reduces to the classical entropy of a wave packet, the Petz-Rényi relative entropy also depends on the symmetric part of the two-point function and is thus genuinely quantum. We also consider the relation with standard subspaces, where we define the Rényi entropy of a vector and show that it admits an upper bound given by the entropy of the vector.