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Law of large numbers and central limit theorem for ergodic quantum processes

Lubashan Pathirana, Jeffrey Schenker

TL;DR

The paper extends the theory of ergodic quantum processes by proving a Law of Large Numbers and a Central Limit Theorem for observables evolving under sequences of positive linear maps sampled along an ergodic dynamics. It develops a rigorous framework based on Perron–Frobenius theory for irreducible maps, a projective geometry on density matrices, and contraction properties to control long-time behavior. The LLN identifies the exponential growth rate $l=\mathbb{E}[\ln \|\varphi_0^*(Z_1)\|]$ governing the asymptotics of $\ln \langle Y, \Phi^{(n)}(X)\rangle$, while the CLT describes Gaussian fluctuations around $nl$ under invertible dynamics and suitable integrability and mixing conditions, with a computable variance $\sigma^2$. The paper also provides mixing conditions that guarantee the CLT hypotheses, linking ergodic theory, operator theory, and quantum processes. Overall, these results quantify both convergence to equilibrium and the fluctuations around it for ergodic quantum processes, with implications for open quantum systems subjected to stochastic noise.

Abstract

A discrete quantum process is represented by a sequence of quantum operations, which are completely positive maps that are not necessarily trace preserving. We consider quantum processes that are obtained by repeated iterations of a quantum operation with noise. Such ergodic quantum processes generalize independent quantum processes. An ergodic theorem describing convergence to equilibrium for a general class of such processes was recently obtained by Movassagh and Schenker. Under irreducibility and mixing conditions, we obtain a central limit type theorem describing fluctuations around the ergodic limit.

Law of large numbers and central limit theorem for ergodic quantum processes

TL;DR

The paper extends the theory of ergodic quantum processes by proving a Law of Large Numbers and a Central Limit Theorem for observables evolving under sequences of positive linear maps sampled along an ergodic dynamics. It develops a rigorous framework based on Perron–Frobenius theory for irreducible maps, a projective geometry on density matrices, and contraction properties to control long-time behavior. The LLN identifies the exponential growth rate governing the asymptotics of , while the CLT describes Gaussian fluctuations around under invertible dynamics and suitable integrability and mixing conditions, with a computable variance . The paper also provides mixing conditions that guarantee the CLT hypotheses, linking ergodic theory, operator theory, and quantum processes. Overall, these results quantify both convergence to equilibrium and the fluctuations around it for ergodic quantum processes, with implications for open quantum systems subjected to stochastic noise.

Abstract

A discrete quantum process is represented by a sequence of quantum operations, which are completely positive maps that are not necessarily trace preserving. We consider quantum processes that are obtained by repeated iterations of a quantum operation with noise. Such ergodic quantum processes generalize independent quantum processes. An ergodic theorem describing convergence to equilibrium for a general class of such processes was recently obtained by Movassagh and Schenker. Under irreducibility and mixing conditions, we obtain a central limit type theorem describing fluctuations around the ergodic limit.
Paper Structure (13 sections, 21 theorems, 125 equations)

This paper contains 13 sections, 21 theorems, 125 equations.

Table of Contents

  1. Introduction and main results
  2. Formal statement of the main results
  3. Positive Linear Maps
  4. Limiting results for eigenmatrices of ergodic quantum processes
  5. Law of Large Numbers
  6. Central Limit Theorem
  7. Background results: geometry of lg, contraction for positive maps, and ergodic arguments
  8. A metric on lg
  9. Contraction Coefficient for
  10. Invertible ergodic dynamics
  11. Proof of the Law of Large Numbers
  12. Proof the Central Limit Theorem
  13. Mixing Conditions

Key Result

Proposition 2.1

If $\phi$ is an irreducible positive map, then there is a unique $R \in \mathbb{S}_D$ such that $\phi(R)= \Lambda R$ for some $\Lambda \in \mathbb{C}$. Furthermore, the eigen-matrix $R$ is non-singular ($R\in \mathbb{S}_D^\circ$) and the eigenvalue $\Lambda=r(\phi) >0$ is the spectral radius of $\ph

Theorems & Definitions (40)

  • Proposition 2.1
  • Definition 1
  • Lemma 2.2: movassagh2022ergodic
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • ...and 30 more