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Quantum Trajectories. Spectral Gap, Quasi-compactness & Limit Theorems

Tristan Benoist, Arnaud Hautecoeur, Clément Pellegrini

TL;DR

This work develops a sharp spectral analysis of the Markov operator $\Pi$ governing quantum trajectories on ${\mathrm P}(\mathbb{C}^k)$ under purification and irreducibility, proving quasi-compactness and a precise peripheral spectrum which yields a robust spectral gap. By constructing analytic perturbations $\Pi_z$ and $\Gamma_z$, the authors derive central limit theorems, Berry-Esseen bounds, and restricted large deviation principles for both empirical averages and the Lyapunov exponent, extending results beyond i.i.d. random matrices to the non-invertible, partially irreducible setting. The approach hinges on a detailed quasi-compact decomposition, explicit cycle-based eigenfunctions, and perturbation theory, enabling sharp fluctuation results for long-time quantum trajectories. The findings provide a solid, transferable framework for long-time behavior of quantum measurement-induced dynamics with rigorous rate estimates and deviation principles, informing both mathematical theory and quantum information applications.

Abstract

Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Inspired by the theory of random products of matrices, it has been shown that these Markov processes admit a unique invariant measure under a purification and an irreducibility assumptions. This paper is devoted to the spectral study of the underlying Markov operator. Using Quasi-compactness, it is shown that this operator admits a spectral gap and the peripheral spectrum is described in a precise manner. Next two perturbations of this operator are studied. This allows to derive limit theorems (Central Limit Theorem, Berry-Esseen bounds and Large Deviation Principle) for the empirical mean of functions of the Markov chain as well as the Lyapounov exponent of the underlying random dynamical system.

Quantum Trajectories. Spectral Gap, Quasi-compactness & Limit Theorems

TL;DR

This work develops a sharp spectral analysis of the Markov operator governing quantum trajectories on under purification and irreducibility, proving quasi-compactness and a precise peripheral spectrum which yields a robust spectral gap. By constructing analytic perturbations and , the authors derive central limit theorems, Berry-Esseen bounds, and restricted large deviation principles for both empirical averages and the Lyapunov exponent, extending results beyond i.i.d. random matrices to the non-invertible, partially irreducible setting. The approach hinges on a detailed quasi-compact decomposition, explicit cycle-based eigenfunctions, and perturbation theory, enabling sharp fluctuation results for long-time quantum trajectories. The findings provide a solid, transferable framework for long-time behavior of quantum measurement-induced dynamics with rigorous rate estimates and deviation principles, informing both mathematical theory and quantum information applications.

Abstract

Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Inspired by the theory of random products of matrices, it has been shown that these Markov processes admit a unique invariant measure under a purification and an irreducibility assumptions. This paper is devoted to the spectral study of the underlying Markov operator. Using Quasi-compactness, it is shown that this operator admits a spectral gap and the peripheral spectrum is described in a precise manner. Next two perturbations of this operator are studied. This allows to derive limit theorems (Central Limit Theorem, Berry-Esseen bounds and Large Deviation Principle) for the empirical mean of functions of the Markov chain as well as the Lyapounov exponent of the underlying random dynamical system.
Paper Structure (21 sections, 27 theorems, 130 equations)

This paper contains 21 sections, 27 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Results
  4. Assumptions
  5. Spectral gap
  6. Tiltings
  7. Limit theorems
  8. Quantum trajectory empirical mean
  9. Lyapunov exponent
  10. Proof of the spectral gap, Theorem \ref{['QCompact']}
  11. Quasi-Compactness
  12. Peripheral spectrum
  13. Proofs for tiltings
  14. Proof of Theorem \ref{['Analytique']}
  15. Proof of Theorem \ref{['AnalyticLog']}
  16. ...and 6 more sections

Key Result

Proposition 3.1

For every $0<\alpha\leq 1$, $\Pi$ is a bounded endomorphism of $C^\alpha({{\mathrm P}({\mathbb C}^k)},{\mathbb C})$.

Theorems & Definitions (52)

  • Proposition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Remark 3.5
  • Theorem 3.6
  • Theorem 3.7: Central Limit Theorem
  • Theorem 3.8: Restricted Large Deviation Principle
  • Theorem 3.9: Central Limit Theorem
  • Theorem 3.10: Restricted Large Deviation Principle
  • ...and 42 more