Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance

Abstract

We study the entropy production of the sandwiched Rényi divergence under the primitive Lindblad equation with Gel'fand-Naimark-Segal-detailed balance. We prove that the Lindblad equation can be identified as the gradient flow of the sandwiched Rényi divergence of any order α ∈ (0, ∞). This extends a previous result by Carlen and Maas [J. Funct. Anal. 273(5), 1810-1869 (2017)] for the quantum relative entropy (i.e., α = 1). Moreover, we show that the sandwiched Rényi divergence of any order α ∈ (0, ∞) decays exponentially fast under the time evolution of such a Lindblad equation.