The Generalized Lyapunov Theorem and its Application to Quantum Channels

Abstract

We give a simple and physically intuitive necessary and sufficient condition for a map acting on a compact metric space to be mixing (i.e. infinitely many applications of the map transfer any input into a fixed convergency point). This is a generalization of the "Lyapunov direct method". First we prove this theorem in topological spaces and for arbitrary continuous maps. Finally we apply our theorem to maps which are relevant in Open Quantum Systems and Quantum Information, namely Quantum Channels. In this context we also discuss the relations between mixing and ergodicity (i.e. the property that there exist only a single input state which is left invariant by a single application of the map) showing that the two are equivalent when the invariant point of the ergodic map is pure.

13 pages, 3 figures

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• Ergodic and Mixing Quantum Channels in Finite Dimensions
• Optimal quantum chain communication by end gates
• The structures of state space concerning Quantum Dynamical Semigroups
• Critical properties of homogeneous binary trees
• On a class of quantum channels, open random walks and recurrence
• Quantum Estimation via Sequential Measurements
• Mediated Homogenization
• Extraction of Singlet States from Noninteracting High-Dimensional Spins
• Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states
• Engineering Stable Discrete-Time Quantum Dynamics via a Canonical QR Decomposition
• Heisenberg Picture Approach to the Stability of Quantum Markov Systems
• Quantum reservoir computing in finite dimensions
• Homogeneous MERA states: an information theoretical analysis
• Efficient Generation of a Maximally Entangled State by Repeated On- and Off-Resonant Scattering of Ancilla Qubits
• Open quantum random walks: ergodicity, hitting times, gambler's ruin and potential theory
• Open Quantum Random Walks and Quantum Markov chains on Trees I: Phase transitions
• Entanglement renormalization and boundary critical phenomena
• Extraction of an Entanglement by Repetition of the Resonant Transmission of an Ancilla Qubit
• Asymptotic Properties for Markovian Dynamics in Quantum Theory and General Probabilistic Theories
• Quantum Compression of Tensor Network States
• Communication Through a Quantum Link
• Information Geometry Approach to Parameter Estimation in Hidden Markov Models
• Ergodic theory of diagonal orthogonal covariant quantum channels
• Necessity of Feedback Control for the Quantum Maxwell Demon in a Finite-Time Steady Feedback Cycle
• Bath Dynamical Decoupling with a Quantum Channel
• A universal constraint for relaxation rates for quantum Markov generators: complete positivity and beyond
• Heating and cooling processes via phaseonium-driven dynamics of cascade systems