Ergodic and Mixing Quantum Channels in Finite Dimensions

Abstract

The paper provides a systematic characterization of quantum ergodic and mixing channels in finite dimensions and a discussion of their structural properties. In particular, we discuss ergodicity in the general case where the fixed point of the channel is not a full-rank (faithful) density matrix. Notably, we show that ergodicity is stable under randomizations, namely that every random mixture of an ergodic channel with a generic channel is still ergodic. In addition, we prove several conditions under which ergodicity can be promoted to the stronger property of mixing. Finally, exploiting a suitable correspondence between quantum channels and generators of quantum dynamical semigroups, we extend our results to the realm of continuous-time quantum evolutions, providing a characterization of ergodic Lindblad generators and showing that they are dense in the set of all possible generators.

replaced with published version

References in corpus (11)

• Full control by locally induced relaxation
• Quantum channels and their entropic characteristics
• Local controllability of quantum networks
• Quantum MERA Channels
• Improved transfer of quantum information using a local memory
• Optimal quantum chain communication by end gates
• Fisher information and asymptotic normality in system identification for quantum Markov chains
• The Generalized Lyapunov Theorem and its Application to Quantum Channels
• Mediated Homogenization
• The inverse eigenvalue problem for quantum channels
• A Protocol For Cooling and Controlling Composite Systems by Local Interactions

Cited by in corpus (64)

• Quantum channels and memory effects
• Quantum collision models: open system dynamics from repeated interactions
• Slow dynamics and thermodynamics of open quantum systems
• Algebraic Theory of Quantum Synchronization and Limit Cycles under Dissipation
• On a class of quantum channels, open random walks and recurrence
• Quantum Estimation via Sequential Measurements
• Artificial quantum thermal bath: Engineering temperature for a many-body quantum system
• Irreducible decompositions and stationary states of quantum channels
• Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states
• Approximate Randomized Benchmarking for Finite Groups
• Quantum reservoir computing in finite dimensions
• Maximum power heat engines and refrigerators in the fast-driving regime
• Entropy Production of Doubly Stochastic Quantum Channels
• Decompositions of Hilbert Spaces, Stability Analysis and Convergence Probabilities for Discrete-Time Quantum Dynamical Semigroups
• Multiplicative Properties of Quantum Channels
• Uniqueness of steady states of Gorini-Kossakowski-Sudarshan-Lindblad equations: a simple proof
• Sandwiched Rényi Convergence for Quantum Evolutions
• Asymptotics of quantum channels
• Open quantum random walks: ergodicity, hitting times, gambler's ruin and potential theory
• Simulation of Quantum Walks and Fast Mixing with Classical Processes
• Eventually entanglement breaking Markovian dynamics: structure and characteristic times
• Theory of Ergodic Quantum Processes
• An ergodic theorem for quantum processes with applications to matrix product states
• Asymptotics of quantum Markov processes: From algebraic structure to characterization of asymptotic states
• Universal quantum uncertainty relations between non-ergodicity and loss of information
• Catalysis always degrades external quantum correlations
• Sequential generalized measurements: Asymptotics, typicality and emergent projective measurements
• Fuzzy measurements and coarse graining in quantum many-body systems
• Theory of Metastability in Discrete-Time Open Quantum Dynamics
• Deviation bounds and concentration inequalities for quantum noises
• Strict Positivity and $D$-Majorization
• Asymptotic Properties for Markovian Dynamics in Quantum Theory and General Probabilistic Theories
• Spectral Properties of Tensor Products of Channels
• Noise-robust exploration of many-body quantum states on near-term quantum devices
• Quantum Compression of Tensor Network States
• Lower bound for simulation cost of open quantum systems: Lipschitz continuity approach
• Exploring Non-Markovianity in Ergodic Channels: Measuring Memory Retention through Ergotropy
• Generalizations of $2$-dimensional diagonal quantum channels with constant Frobenius norm
• Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance
• Information Geometry Approach to Parameter Estimation in Hidden Markov Models
• A resource theoretical unification of Mpemba effects: classical and quantum
• Faithfulness and sensitivity for ancilla-assisted process tomography
• Application of Shemesh theorem to quantum channels
• How coherence measurements of a qubit steer its quantum environment
• Ergodic theory of diagonal orthogonal covariant quantum channels
• Necessity of Feedback Control for the Quantum Maxwell Demon in a Finite-Time Steady Feedback Cycle
• Peripherally automorphic unital completely positive maps
• Reachability in Controlled Markovian Quantum Systems: An Operator-Theoretic Approach
• Input-dependence in quantum reservoir computing
• Almost All Quantum Channels Are Diagonalizable
• Perfect Sampling for Quantum Gibbs States
• Emergent non-Markovianity and dynamical quantification of the quantum switch
• Spin-chain based quantum thermal machines
• Repeated interaction processes in the continuous-time limit, applied to quadratic fermionic systems
• Heisenberg-Limited Quantum Metrology without Ancillae
• Bath Dynamical Decoupling with a Quantum Channel
• Channel-based framework for phase esimation of multiple eigenvalues
• Spectral anomalies and broken symmetries in maximally chaotic quantum maps
• Transitions of the Lyapunov spectrum and entanglement entropy in monitored quantum dynamics with homogeneous unitary gates
• Nonhomogeneous quantum Markov chains and a notion of ergodicity
• Constructing k-local parent Lindbladians for matrix product density operators
• Zero-error communication under discrete-time Markovian dynamics
• Entanglement structure for finite system under dual-unitary dynamics
• Heating and cooling processes via phaseonium-driven dynamics of cascade systems