Randomized Quantum Hamiltonian Systems

TL;DR

This paper introduces a method for averaging random quantum transformations, generalizing convergence concepts, and applies these results to analyze the dynamics of quantum systems with random Hamiltonians.

Contribution

It generalizes weak convergence for random unitary and self-adjoint groups and applies this to quantum system dynamics with random quantization.

Findings

Convergence results align with the central limit theorem for independent random vectors.

Provides a framework for analyzing quantum systems with random Hamiltonian dynamics.

Extends classical probability concepts to quantum transformation sequences.

Abstract

We present a procedure for averaging one-parameter random unitary groups and random self-adjoint groups. Central to this is a generalization of the notion of weak convergence of a sequence of measures and the corresponding generalization of the concept of convergence in distribution. The convergence is established in determination of the sequence of compositions of independent random transformations. When sequences of compositions of independent random transformations of the shift by the Euclidean vector in space, the results obtained coincide with the central limit theorem for the sums independent random vectors. The results are applied to the dynamics of quantum systems arising random quantization of the classical Hamiltonian system.