Irreducibility of Quantum Markov Semigroups, uniqueness of invariant states and related properties

2025/12/12 by Fagnola, Franco, Girotti, Federico · 1 citation
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Quantum many-body systems #Markov Chains and Monte Carlo Methods

paper · doi:10.48550/arxiv.2512.11517

Abstract

We present different characterizations of the notion of irreducibility for Quantum Markov Semigroups (QMSs) and investigate its relationship with other relevant features of the dynamics, such as primitivity, positivity improvement and relaxation; in particular, we show that irreducibility, primitivity and relaxation towards a faithful invariant density are equivalent when the semigroup admits an invariant density. Moreover, in the case of uniformly continuous QMSs, we present several useful ways of checking irreducibility in terms of the operators appearing in the generator in GKLS form. Our exposition is as much self-contained as possible, we present some well known results with elementary proofs (collecting all the relevant literature) and we derive new ones. We study both finite and infinite dimensional evolutions and we remark that many results only require the QMS to be made of Schwarz maps.

Citations

• Asymptotic dynamics in the Heisenberg picture: attractor subspace and Choi-Effros product
• State Space Decomposition of Quantum Dynamical Semigroups
• Decoherence-free algebras in quantum dynamics
• Peripheral Poisson Boundary
• Absorption and Fixed Points for Semigroups of Quantum Channels
• The general structure of the Decoherence-free subalgebra for uniformly continuous Quantum Markov semigroups
• Invariant measure for stochastic Schrödinger equations
• An Ergodic Theorem for Quantum Counting Processes
• Completely positive dynamical semigroups of <i>N</i>-level systems

Cited by

• On the existence of the KMS spectral gap in Gaussian quantum Markov semigroups

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• Exponential Relative Entropy Decay Along Quantum Markov Semigroups