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We study the entropy production of the sandwiched R\'enyi divergence underthe primitive Lindblad equation with GNS-detailed balance. We prove that theLindblad equation can be identified as the gradient flow of the sandwichedR\'enyi divergence of any order ${\alpha} \in (0, \infty)$. This extends aprevious 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 R\'enyi divergence of any order ${\alpha} \in (0,\infty)$ decays exponentially fast under the time-evolution of such a Lindbladequation.

Gradient flow structure and exponential decay of the sandwiched R\'enyi divergence for primitive Lindblad equations with GNS-detailed balance