Sandwiched Rényi Convergence for Quantum Evolutions

Abstract

We study the speed of convergence of a primitive quantum time evolution towards its fixed point in the distance of sandwiched Rényi divergences. For each of these distance measures the convergence is typically exponentially fast and the best exponent is given by a constant (similar to a logarithmic Sobolev constant) depending only on the generator of the time evolution. We establish relations between these constants and the logarithmic Sobolev constants as well as the spectral gap. An important consequence of these relations is the derivation of mixing time bounds for time evolutions directly from logarithmic Sobolev inequalities without relying on notions like lp-regularity. We also derive strong converse bounds for the classical capacity of a quantum time evolution and apply these to obtain bounds on the classical capacity of some examples, including stabilizer Hamiltonians under thermal noise.

35 pages, 4 figures. Version to be published in the Quantum Journal

References in corpus (7)

• Coding Theorem and Strong Converse for Quantum Channels
• On thermalization in Kitaev's 2D model
• Hypercontractivity of quasi-free quantum semigroups
• Relative Entropy Convergence for Depolarizing Channels
• Entropy Production of Doubly Stochastic Quantum Channels
• Area law for fixed points of rapidly mixing dissipative quantum systems
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• Relative entropy for von Neumann subalgebras
• Exponential decay of Rényi divergence under Fokker-Planck equations
• Eventually entanglement breaking Markovian dynamics: structure and characteristic times
• Noise-robust exploration of many-body quantum states on near-term quantum devices
• Matrix Poincaré, Φ-Sobolev inequalities, and quantum ensembles
• Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance
• Limitations of variational quantum algorithms: a quantum optimal transport approach
• Information storage and transmission under Markovian noise
• A Randomized Method for Simulating Lindblad Equations and Thermal State Preparation