The law of large numbers for discrete generalized quantum channels
S. V. Dzhenzher, V. Zh. Sakbaev
TL;DR
The paper addresses the problem of establishing a law of large numbers for compositions of i.i.d. random pre-channels on Schatten classes $\mathcal{T}_p$ with $p\in[1,2]$, extending the known $p=2$ result. It develops a discrete-measurability framework and uses Chebyshev-type estimates, adjoint and super-operator techniques, and independence properties to prove almost-sure convergence in the strong operator topology on $\mathcal{T}_2$ for the product process $W_n(t)=e^{A_1 t/n}\cdots e^{A_n t/n}$ to the deterministic semigroup $e^{(\mathbb{E}A)t}$, uniformly on compact time intervals; a conjectured extension to $\mathcal{T}_p$ is discussed. The work provides a non-commutative LLN for random quantum dynamics and random generalized quantum channels, leveraging the discretized setting and a decomposition of the product into mean and fluctuation terms. These results contribute to rigorous understanding of random quantum evolutions and open quantum systems in Banach spaces, with potential impact on the study of random quantum processes and channels.
Abstract
We consider random linear operators $Ω\to \mathcal{L}(\mathcal{T}_p, \mathcal{T}_p)$ acting in a $p$-th Schatten class $\mathcal{T}_p$ in a separable Hilbert space $\mathcal{H}$ for some $1 \leqslant p < \infty$. Such a superoperator is called a pre-channel since it is an extension of a quantum channel to a wider class of operators without requirements of trace-preserving and positivity. Instead of the sum of i.i.d. variables there may be considered the composition of random semigroups $e^{A_i t/n}$ in the Banach space $\mathcal{T}_p$. The law of large numbers is known in the case $p=2$ in the form of the usual law of large numbers for random operators in a Hilbert space. We obtain the law of large numbers for the case $1\leqslant p \leqslant 2$.