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Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance. (English) Zbl 1414.81141

Summary: We study the entropy production of the sandwiched Rényi divergence under the primitive Lindblad equation with Gel’fand-Naimark-Segal-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 E. A. Carlen and J. Maas [J. Funct. Anal. 273, No. 5, 1810–1869 (2017; Zbl 1386.46057)] 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.
©2019 American Institute of Physics

MSC:

81S22 Open systems, reduced dynamics, master equations, decoherence
94A17 Measures of information, entropy
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
47D07 Markov semigroups and applications to diffusion processes

Citations:

Zbl 1386.46057
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References:

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