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 GNS-detailed balance. We prove that the Lindblad equation can be identified as the gradient flow of the sandwiched Rényi divergence of any order $\alpha \in (0,\infty )$. This extends a previous result by Carlen and Maas [Journal of Functional Analysis, 273(5), 1810-1869] for the quantum relative entropy (i.e., $\alpha =1$). Moreover, we show that the sandwiched Rényi divergence of any order $\alpha \in (0,\infty )$ decays exponentially fast under the time-evolution of such a Lindblad equation.

43 pages; 2 figures; add a new section about the necessary condition of having a gradient flow structure