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Central Limit Theorems for Outcome Records in Disordered Quantum Trajectories

Lubashan Pathirana

Abstract

We prove annealed central limit theorems for finite pattern counts in the measurement record of discrete-time quantum trajectories generated by repeated measurements in a disordered environment. Under summable mixing assumptions on the environment and an annealed trace-norm forgetting property for the associated non-selective channel cocycle, we first establish the CLT under the annealed law determined by the dynamically stationary state. This part applies to general disordered quantum instruments and, in particular, is not restricted to the perfect-measurement regime; it complements both the corresponding law of large numbers for disordered measurement records and the homogeneous central limit theorem. We then introduce a coupling-based notion of admissibility for initial states and show that the same Gaussian limit extends to every admissible initial law, with unchanged centering and asymptotic variance. In the perfect-measurement setting, we further identify a general condition ensuring admissibility for every initial state, and hence obtain a universal annealed central limit theorem. We also provide practical sufficient criteria for this condition and verify the assumptions across a broad family of examples, including disordered walk-type models generated by finite group actions

Central Limit Theorems for Outcome Records in Disordered Quantum Trajectories

Abstract

We prove annealed central limit theorems for finite pattern counts in the measurement record of discrete-time quantum trajectories generated by repeated measurements in a disordered environment. Under summable mixing assumptions on the environment and an annealed trace-norm forgetting property for the associated non-selective channel cocycle, we first establish the CLT under the annealed law determined by the dynamically stationary state. This part applies to general disordered quantum instruments and, in particular, is not restricted to the perfect-measurement regime; it complements both the corresponding law of large numbers for disordered measurement records and the homogeneous central limit theorem. We then introduce a coupling-based notion of admissibility for initial states and show that the same Gaussian limit extends to every admissible initial law, with unchanged centering and asymptotic variance. In the perfect-measurement setting, we further identify a general condition ensuring admissibility for every initial state, and hence obtain a universal annealed central limit theorem. We also provide practical sufficient criteria for this condition and verify the assumptions across a broad family of examples, including disordered walk-type models generated by finite group actions

Paper Structure

This paper contains 38 sections, 31 theorems, 760 equations.

Table of Contents

  1. Introduction
  2. Disordered Quantum Trajectories
  3. Outcome Frequencies.
  4. Dynamically Stationary State.
  5. Assumptions and Main Result
  6. Related Literature
  7. Examples
  8. Non-disordered Examples
  9. Environment Examples
  10. I.i.d. Instrument Environments.
  11. Markov-driven Instrument Environments.
  12. Deterministic Instruments
  13. Disordered Instrument Examples
  14. Disordered Walk-type Measurements
  15. Preliminaries and Notation
  16. ...and 23 more sections

Key Result

Theorem 1.1

Assume that $\left(\Omega,\mcF,\mathop{\mathrm{\mathsf{pr}}}\nolimits,\theta\right)$ is an invertible, ergodic, $\mathop{\mathrm{\mathsf{pr}}}\nolimits$-preserving dynamical system, and let $\omega\mapsto\mcV_\omega$ be a random instrument on a finite outcome set $\mcA$. Suppose there exists a uniqu

Theorems & Definitions (73)

  • Theorem 1.1: LLN for Outcome Frequencies qtlln
  • Remark 1.1
  • Theorem 1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2: Annealed CLT for ()
  • Definition 1: Admissible initial state relative to a reference law
  • Proposition 2
  • Theorem 3: Transfer CLT for perfect measurements
  • Proposition 3
  • ...and 63 more