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

Journal Article

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

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### Duke Authors

### Cited Authors

- Cao, Y; Lu, J; Lu, Y