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

Abstract

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

arXiv comment43 pages; 2 figures; add a new section about the necessary condition of having a gradient flow structure

References in corpus (29)

• On quantum Renyi entropies: a new generalization and some properties
• The second laws of quantum thermodynamics
• Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
• Sandwiched Rényi Divergence Satisfies Data Processing Inequality
• Monotonicity of a relative Rényi entropy
• Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies
• alpha-z-relative Renyi entropies
• Quantum logarithmic Sobolev inequalities and rapid mixing
• On the Mean Field and Classical Limits of Quantum Mechanics
• The ${\chi }^2$-divergence and Mixing times of quantum Markov processes
• Ergodic and Mixing Quantum Channels in Finite Dimensions
• Monotonicity of the Quantum Relative Entropy Under Positive Maps
• Generators of Detailed Balance Quantum Markov Semigroups
• Monge Metric on the Sphere and Geometry of Quantum States
• On Variational Expressions for Quantum Relative Entropies
• Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems
• Quantum jumps and entropy production
• Generators of KMS Symmetric Markov Semigroups on B(h) Symmetry and Quantum Detailed Balance
• Monge Distance between Quantum States
• Relative Entropy Convergence for Depolarizing Channels
• Wave Packets and the Quadratic Monge-Kantorovich Distance in Quantum Mechanics
• Inequalities for quantum divergences and the Audenaert-Datta conjecture
• Entropy Production for Quantum Markov Semigroups
• Entropy Production of Doubly Stochastic Quantum Channels
• Sandwiched Rényi Convergence for Quantum Evolutions
• Exponential decay of Rényi divergence under Fokker-Planck equations
• A Noncommutative Transport Metric and Symmetric Quantum Markov Semigroups as Gradient Flows of the Entropy
• On the Matrix Monge-Kantorovich Problem
• $L^2$-Wasserstein distances of tracial $W^{\ast }$-algebras and their disintegration problem

Cited by in corpus (5)

• Relative entropy for von Neumann subalgebras
• Exponential decay of Rényi divergence under Fokker-Planck equations
• A resource theoretical unification of Mpemba effects: classical and quantum
• Lindblad evolution as gradient flow
• Double-Bracket Master Equations: Phase-Space Representation and Classical Limit