Quantum Markov Semigroups and Information Dynamics

Quantum Markov semigroups (QMS) generalise classical Markov processes to open quantum systems, capturing their continuous-time evolution under both coherent Hamiltonian dynamics and dissipative interactions with an environment. A QMS is defined by a family of completely positive, trace-preserving maps whose generator comprises unitary and dissipative parts satisfying the Lindblad form. Central to the field are questions of convergence to equilibrium, spectral gap estimates and entropy production, which underpin thermalisation processes in quantum thermodynamics and provide benchmarks for the stability of quantum information protocols. Information-theoretic quantities such as relative entropy and quantum Fisher information interplay with functional inequalities—most notably logarithmic Sobolev and Poincaré inequalities—to quantify mixing times and concentration of measure in quantum ensembles. In parallel, methods from optimal transport have been extended into the non-commutative realm, introducing quantum Wasserstein distances and curvature lower bounds. These developments have enriched our understanding of decoherence, resource distribution in many-body systems and error suppression in quantum computing architectures.

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Technical terms

Quantum Markov semigroup: A family of completely positive, trace-preserving maps describing the time evolution of an open quantum system in continuous time.

Lindblad generator: The operator governing a quantum Markov semigroup, comprising Hamiltonian and dissipative contributions that ensure complete positivity.

Logarithmic Sobolev inequality: A functional inequality relating entropy decay to the Dirichlet form of a generator, used to bound convergence rates of quantum dynamics.

Wasserstein distance (quantum): A non-commutative generalisation of the classical transport distance, measuring the minimal cost of transforming one quantum state into another.

Relative entropy: A measure of distinguishability between two quantum states, fundamental for quantifying convergence to equilibrium and information loss.

Quantum Fisher information: A metric that quantifies the sensitivity of a quantum state to infinitesimal parameter changes, central to quantum estimation theory.

Ricci curvature bound (quantum): A lower bound on a quantum analogue of Ricci curvature, unifying transport and functional inequalities in non-commutative geometry.

References

1. Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems. Journal of Statistical Physics (2019).
2. Relating Relative Entropy, Optimal Transport and Fisher Information: A Quantum HWI Inequality. Annales Henri Poincaré (2020).
3. Complete Entropic Inequalities for Quantum Markov Chains. Archive for Rational Mechanics and Analysis (2022).
4. Complete Gradient Estimates of Quantum Markov Semigroups. Communications in Mathematical Physics (2021).
5. Quantum Concentration Inequalities. Annales Henri Poincaré (2022).