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Limit theorems for the evolution of quantum pure states

S. V. Dzhenzher, V. Zh. Sakbaev

TL;DR

This work analyzes limit behavior of quantum states under compositions of i.i.d. random unitary channels, focusing on strong law and central limit type results. It develops two complementary formalisms: coordinate/kernel representations and convergence in the Weak Operator Topology, leveraging shift-operator dynamics and the Fourier transform. The main contributions are SLLN and CLT results for kernels $\rho[S_{\xi_1/n}\cdots S_{\xi_n/n}u]$ and $\rho[S_{\xi_1/\sqrt{n}}\cdots S_{\xi_n/\sqrt{n}}u]$, with Gaussian limits governed by $\mu$ and $\Sigma$, respectively, along with WOT-based analogues and random-walk/Brownian limits; the results extend to impulse transforms as well. These findings advance understanding of random quantum dynamical semigroups and open pathways for generalizations to other measures, infinite-dimensional spaces, and non-pure states in quantum dynamics.

Abstract

Limit theorems of strong law of large numbers and central limit theorem types are obtained for the compositions of independent identically distributed random unitary channels.

Limit theorems for the evolution of quantum pure states

TL;DR

This work analyzes limit behavior of quantum states under compositions of i.i.d. random unitary channels, focusing on strong law and central limit type results. It develops two complementary formalisms: coordinate/kernel representations and convergence in the Weak Operator Topology, leveraging shift-operator dynamics and the Fourier transform. The main contributions are SLLN and CLT results for kernels and , with Gaussian limits governed by and , respectively, along with WOT-based analogues and random-walk/Brownian limits; the results extend to impulse transforms as well. These findings advance understanding of random quantum dynamical semigroups and open pathways for generalizations to other measures, infinite-dimensional spaces, and non-pure states in quantum dynamics.

Abstract

Limit theorems of strong law of large numbers and central limit theorem types are obtained for the compositions of independent identically distributed random unitary channels.
Paper Structure (5 sections, 11 theorems, 29 equations)

This paper contains 5 sections, 11 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Basic notions
  3. Coordinate representation
  4. Weak Operator Topology of quantum states
  5. Conclusions

Key Result

Theorem 3.1

Let $\ket{u}\in \mathcal{H}$. Let $\{\xi_n\}$ be i.i.d. random vectors with finite expectation $\mu\in\mathbb{R}^d$. Then for any $x,y \in \mathbb{R}^d$

Theorems & Definitions (17)

  • Theorem 3.1: SLLN for kernel
  • Proposition 3.3
  • proof
  • Theorem 3.4: Continuous Mapping Billingsley2013convergence
  • proof : Proof of Theorem \ref{['t:slln-kernel']}
  • proof : Proof of Theorem \ref{['t:clt-kernel']}
  • Theorem 4.1: SLLN in WOT
  • Theorem 4.2: CLT in WOT
  • Theorem 4.3: Random walk
  • Lemma 4.4: AmosovSakbaev
  • ...and 7 more