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

We study the entropy production of the sandwiched Renyi 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 Renyi 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 Renyi divergence of any order ${alpha} in (0, infty)$ decays exponentially fast under the time-evolution of such a Lindblad equation.