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

Yu Cao; Jianfeng Lu; Yulong Lu

doi:10.1063/1.5083065

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

Yu Cao, Jianfeng Lu, Yulong Lu

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Abstract

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 α ∈ (0, ∞). This extends a previous result by Carlen and Maas [J. Funct. Anal. 273(5), 1810-1869 (2017)] for the quantum relative entropy (i.e., α = 1). Moreover, we show that the sandwiched Rényi divergence of any order α ∈ (0, ∞) decays exponentially fast under the time evolution of such a Lindblad equation.

Original language	English (US)
Article number	052202
Journal	Journal of Mathematical Physics
Volume	60
Issue number	5
DOIs	https://doi.org/10.1063/1.5083065
State	Published - May 1 2019
Externally published	Yes

Bibliographical note

Funding Information:
This work was partially supported by the National Science Foundation under Grant No. DMS-1454939. We thank Iman Marvian and Henry Pfister for helpful discussions.

Publisher Copyright:
© 2019 Author(s).

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10.1063/1.5083065

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title = "Gradient flow structure and exponential decay of the sandwiched R{\'e}nyi divergence for primitive Lindblad equations with GNS-detailed balance",

abstract = "We study the entropy production of the sandwiched R{\'e}nyi 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 R{\'e}nyi divergence of any order α ∈ (0, ∞). This extends a previous result by Carlen and Maas [J. Funct. Anal. 273(5), 1810-1869 (2017)] for the quantum relative entropy (i.e., α = 1). Moreover, we show that the sandwiched R{\'e}nyi divergence of any order α ∈ (0, ∞) decays exponentially fast under the time evolution of such a Lindblad equation.",

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AU - Lu, Yulong

N1 - Funding Information: This work was partially supported by the National Science Foundation under Grant No. DMS-1454939. We thank Iman Marvian and Henry Pfister for helpful discussions. Publisher Copyright: © 2019 Author(s).

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N2 - 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 α ∈ (0, ∞). This extends a previous result by Carlen and Maas [J. Funct. Anal. 273(5), 1810-1869 (2017)] for the quantum relative entropy (i.e., α = 1). Moreover, we show that the sandwiched Rényi divergence of any order α ∈ (0, ∞) decays exponentially fast under the time evolution of such a Lindblad equation.

AB - 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 α ∈ (0, ∞). This extends a previous result by Carlen and Maas [J. Funct. Anal. 273(5), 1810-1869 (2017)] for the quantum relative entropy (i.e., α = 1). Moreover, we show that the sandwiched Rényi divergence of any order α ∈ (0, ∞) decays exponentially fast under the time evolution of such a Lindblad equation.

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

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