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

TL;DR

This paper demonstrates that primitive Lindblad equations with GNS-detailed balance can be viewed as gradient flows of the sandwiched R\'enyi divergence, which decays exponentially over time, extending previous results for quantum relative entropy.

Contribution

It generalizes the gradient flow structure of Lindblad equations to all sandwiched R\'enyi divergences, not just the quantum relative entropy, and proves exponential decay for all orders.

Findings

Lindblad equations are gradient flows of sandwiched R\'enyi divergence.

Exponential decay of sandwiched R\'enyi divergence for all \( \alpha > 0 \).

Extension of previous results from quantum relative entropy to all R\'enyi divergences.

Abstract

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.