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Quenched large deviations of Birkhoff sums along random quantum measurements

Renaud Raquépas, Jeffrey Schenker

TL;DR

This work establishes a quenched large deviation principle for Birkhoff-like sums built from random quantum measurements driven by an ergodic process. By constructing analytic deformations of the instrument maps and analyzing the associated top Lyapunov exponent \lambda(\alpha) through the Gärtner–Ellis framework, the authors obtain a rate function \lambda^*(s) governing fluctuations that hold for almost every environment realization. The approach enables a robust treatment beyond Markovian randomness and yields differentiability and regularity of the cumulant generating function, with a key application to entropy production in the two-time measurement framework. Under a time-reversal invariance (TRI) symmetry, the paper proves a Gallavotti–Cohen type symmetry for the entropy-production rate function and proves that the Clausius-type and information-theoretic notion of entropy production share the same large deviation behavior, deepening the thermodynamic interpretation of quantum measurement processes.

Abstract

We prove a quenched version of the large deviation principle for Birkhoff-like sums along a sequence of random quantum measurements driven by an ergodic process. We apply the result to the study of entropy production in the two-time measurement framework.

Quenched large deviations of Birkhoff sums along random quantum measurements

TL;DR

This work establishes a quenched large deviation principle for Birkhoff-like sums built from random quantum measurements driven by an ergodic process. By constructing analytic deformations of the instrument maps and analyzing the associated top Lyapunov exponent \lambda(\alpha) through the Gärtner–Ellis framework, the authors obtain a rate function \lambda^*(s) governing fluctuations that hold for almost every environment realization. The approach enables a robust treatment beyond Markovian randomness and yields differentiability and regularity of the cumulant generating function, with a key application to entropy production in the two-time measurement framework. Under a time-reversal invariance (TRI) symmetry, the paper proves a Gallavotti–Cohen type symmetry for the entropy-production rate function and proves that the Clausius-type and information-theoretic notion of entropy production share the same large deviation behavior, deepening the thermodynamic interpretation of quantum measurement processes.

Abstract

We prove a quenched version of the large deviation principle for Birkhoff-like sums along a sequence of random quantum measurements driven by an ergodic process. We apply the result to the study of entropy production in the two-time measurement framework.

Paper Structure

This paper contains 13 sections, 19 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Setup and main result
  3. Application to entropy production
  4. Proof of Theorem \ref{['thm:main-abstract']}
  5. Notation and preliminary lemmas
  6. Reduction appealing to the Gärtner--Ellis theorem
  7. Simultaneous existence of limiting objects
  8. Regularity of the limiting objects
  9. Proofs for Section \ref{['sec:EP']}
  10. Abstract measure-theoretic results
  11. Almost sure convergence of convex functions
  12. A sub-additive ergodic upper bound for continuous families
  13. Vanishing of ergodic differences

Key Result

Theorem 2.3

If A1, A2 and A3 hold and $\lambda^*(s) = \sup_{\alpha\in{\mathbb R}} (s\alpha - \lambda(\alpha))$ is the Legendre transform of the Lyapunov exponents $\lambda(\alpha)$ in eq:lambda-alpha, then, for $\mathbb{P}$-almost all $\omega$, we have for every Borel set $E\subseteq{\mathbb R}$.

Theorems & Definitions (48)

  • Example 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Example 2.6
  • Example 3.1
  • Remark 3.2
  • Lemma 3.4
  • Lemma 3.5
  • ...and 38 more