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Ergodic Theorems for Quantum Trajectories under Disordered Generalized Measurements

Owen Ekblad, Eloy Moreno-Nadales, Lubashan Pathirana

TL;DR

The paper analyzes disordered quantum trajectories arising from repeated generalized measurements in a random environment, formulating them as Markov chains in random environments (MCRE) driven by random Kraus ensembles. Under an invertible, ergodic base dynamics and a dynamically ergodic disorder with a unique stationary state $ ho_{ss}$, it proves annealed and quenched ergodic theorems, yielding a strong law of large numbers for measurement outcomes and enabling computation of block-outcome frequencies from the equilibrium state. The framework accommodates various disorder structures (i.i.d., Markovian, periodic, quasiperiodic, and constant) and extends the earlier noise-free results of Kummerer and Maassen to disordered quantum trajectories. Methodologically, the work employs random Kraus ensembles, quenched and annealed quantum measures, skew-product dynamics, and ergodic theorems (Beck–Schwartz) to derive LLNs and ergodic properties for both the annealed and quenched settings. These results illuminate the long-time statistical behavior of quantum trajectories in disordered environments, with potential implications for quantum optics experiments and the theory of open quantum systems.

Abstract

We consider quantum trajectories arising from disordered, repeated generalized measurements, which have the structure of Markov chains in random environments (MCRE) with dynamically-defined transition probabilities; we call these disordered quantum trajectories. Under the assumption that the underlying disordered open quantum dynamical system approaches a unique equilibrium in time averages, we establish a strong law of large numbers for measurement outcomes arising from disordered quantum trajectories, which follows after we establish general annealed ergodic theorems for the corresponding MCRE. The type of disorder our model allows includes the random settings where the disorder is i.i.d. or Markovian, the periodic (resp. quasiperiodic) settings where the disorder has periodic (resp. quasiperiodic) structure, and the nonrandom setting where the disorder is constant through time. In particular, our work extends the earlier noise-free results of Kümmerer and Maassen to the present disordered framework.

Ergodic Theorems for Quantum Trajectories under Disordered Generalized Measurements

TL;DR

The paper analyzes disordered quantum trajectories arising from repeated generalized measurements in a random environment, formulating them as Markov chains in random environments (MCRE) driven by random Kraus ensembles. Under an invertible, ergodic base dynamics and a dynamically ergodic disorder with a unique stationary state , it proves annealed and quenched ergodic theorems, yielding a strong law of large numbers for measurement outcomes and enabling computation of block-outcome frequencies from the equilibrium state. The framework accommodates various disorder structures (i.i.d., Markovian, periodic, quasiperiodic, and constant) and extends the earlier noise-free results of Kummerer and Maassen to disordered quantum trajectories. Methodologically, the work employs random Kraus ensembles, quenched and annealed quantum measures, skew-product dynamics, and ergodic theorems (Beck–Schwartz) to derive LLNs and ergodic properties for both the annealed and quenched settings. These results illuminate the long-time statistical behavior of quantum trajectories in disordered environments, with potential implications for quantum optics experiments and the theory of open quantum systems.

Abstract

We consider quantum trajectories arising from disordered, repeated generalized measurements, which have the structure of Markov chains in random environments (MCRE) with dynamically-defined transition probabilities; we call these disordered quantum trajectories. Under the assumption that the underlying disordered open quantum dynamical system approaches a unique equilibrium in time averages, we establish a strong law of large numbers for measurement outcomes arising from disordered quantum trajectories, which follows after we establish general annealed ergodic theorems for the corresponding MCRE. The type of disorder our model allows includes the random settings where the disorder is i.i.d. or Markovian, the periodic (resp. quasiperiodic) settings where the disorder has periodic (resp. quasiperiodic) structure, and the nonrandom setting where the disorder is constant through time. In particular, our work extends the earlier noise-free results of Kümmerer and Maassen to the present disordered framework.

Paper Structure

This paper contains 8 sections, 18 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Related literature
  4. Notations and preliminary results
  5. Disordered quantum trajectories and related concepts
  6. Associated probability measures
  7. Technical results and proofs of theorems
  8. Proofs of theorems

Key Result

Theorem 1

Assume that $\left(\Omega, \mcF, \mathop{\mathrm{Pr}}\nolimits, \theta\right)$ is an invertible, ergodic $\mathop{\mathrm{Pr}}\nolimits$-preserving dynamical system, and let $\omega\mapsto \mcV_\omega$ define a random Kraus ensemble indexed by $\mcA$. If $\mcV$ satisfies Dyn-Erg with unique stochast holds for $\mbQ_{\vartheta; \omega}$-almost every $\bar{a}\in\mathop{\mathrm{\mcA^\mbN}}\nolimits$.

Theorems & Definitions (33)

  • Theorem 1: Law of Large Numbers for Measurement Outcomes of Disordered Quantum Trajectories
  • Theorem 2: Annealed Ergodic Theorem
  • Theorem 3: Law of Large Numbers for Annealed Quantum Probability
  • Theorem 4: Quenched Ergodic Theorem
  • Definition 1: Dynamical Ergodicity
  • Proposition 2.1
  • Definition 2: Quenched Quantum Probability
  • Lemma 2.2
  • proof
  • Definition 3: Matrix-valued Quenched Quantum Probability
  • ...and 23 more