The Strong Law of Large Numbers for random semigroups with unbounded generators on uniformly smooth Banach spaces

TL;DR

This work addresses the Strong Law of Large Numbers for random semigroups with unbounded generators acting on uniformly smooth Banach spaces, with applications to random open quantum dynamics. The authors combine Chernoff product formula techniques with a Burkholder-type martingale bound tailored to $p$-smooth Banach spaces to control fluctuations of products of random semigroups. They prove that the random products $e^{-L_1 t/n}\dots e^{-L_n t/n}$ converge almost surely in the strong operator topology to the deterministic semigroup $e^{-L_0 t}$, uniformly on compact $t$-intervals, and illustrate the result via a quantum depolarizing-channel example $D_{\xi_1/n}\cdots D_{\xi_n/n} \to D_{1-\mathbb{E}\xi_1}$. This generalizes LLN-type limit results beyond Hilbert spaces and provides new tools for analyzing random dynamics in Banach spaces, including open quantum systems.

Abstract

We consider random linear unbounded operators $Ω\to \mathcal{L}(\mathcal{X}, \mathcal{X})$ on a Banach space $\mathcal{X}$. For example, such random operators may be random quantum channels. The Law of Large Numbers is known when $\mathcal{X}$ is a Hilbert space, in the form of the usual Law of Large Numbers for random operators, and in some other particular cases. Instead of the sum of i.i.d. variables, there may be considered the composition of random semigroups $e^{A_i t/n}$. We obtain the Strong Law of Large Numbers in Strong Operator Topology for random semigroups of unbounded linear operators on a uniformly smooth Banach space.

Theorems & Definitions (9)

• Theorem 2.1
• Theorem 2.2
• proof
• Theorem 3.1: Chernoff Product Formula
• proof
• Lemma 3.2
• proof
• Theorem 3.3: Burkholder-type
• proof : Proof of Theorem \ref{['t:slln-unbound']}