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Ergodic Theory of Inhomogeneous Quantum Processes

Abdessatar Souissi

Abstract

This work develops a rigorous framework for analysing ergodicity and mixing in time-inhomogeneous quantum dynamics. It considers quantum evolutions generated by sequences of quantum channels and examines in detail the relationship between the forward and backward dynamics, showing that they are generically nonequivalent in a structurally meaningful way. A central contribution is the adoption of a quantum Markov-Dobrushin approach to quantify mixing, which yields sharpened conditions for convergence rates and for establishing exponential stability of the induced dynamics. The resulting formalism not only extends classical and stationary quantum theories, but also naturally accommodates non-translationally invariant matrix product states, thereby providing a unified interface with experimentally relevant quantum many-body systems.

Ergodic Theory of Inhomogeneous Quantum Processes

Abstract

This work develops a rigorous framework for analysing ergodicity and mixing in time-inhomogeneous quantum dynamics. It considers quantum evolutions generated by sequences of quantum channels and examines in detail the relationship between the forward and backward dynamics, showing that they are generically nonequivalent in a structurally meaningful way. A central contribution is the adoption of a quantum Markov-Dobrushin approach to quantify mixing, which yields sharpened conditions for convergence rates and for establishing exponential stability of the induced dynamics. The resulting formalism not only extends classical and stationary quantum theories, but also naturally accommodates non-translationally invariant matrix product states, thereby providing a unified interface with experimentally relevant quantum many-body systems.

Paper Structure

This paper contains 13 sections, 10 theorems, 152 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Core Contributions
  3. Application to Inhomogeneous Matrix Product States
  4. Related Studies
  5. Preliminaries
  6. Ergodicity and Mixing for Inhomogeneous Quantum Processes
  7. Ergodicity of Inhomogeneous Markov Chains
  8. Forward and backward quantum Dynamics
  9. Weakly Mixing Inhomogeneous Quantum processes
  10. Trajectory-Ergodic Characterization of Weak Mixing
  11. A Markov-Dobrushin Mixing Condition
  12. Mixing Properties for Inhomogeneous Matrix Product States
  13. Discussion and outlook

Key Result

Theorem 1.1

Let $\{\Phi_n\}_{n \in \mathbb{N}}$ be a sequence of quantum channels on $\mathfrak{S}(\mathcal{H})$. Then the following statements hold:

Figures (3)

  • Figure 1: Mixing and Ergodicity Hierarchy for Inhomogeneous Quantum Processes in Finite Dimension
  • Figure 2: Hierarchy of inclusion relationships among classes of homogeneous quantum processes on $\mathfrak{S}(\mathcal{H})$:
  • Figure 3: Hierarchy of the inclusion relationships among sets of weakly ergodic inhomogeneous quantum processes on $\mathfrak{S}(\mathcal{H})$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Convergence of inhomogeneous matrix product states
  • Theorem 2.1: Kraus Representation
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 3.1
  • Remark 3.2
  • Definition 3.3: Ergodicity Hierarchy with Averages
  • ...and 29 more