Banach and counting measures, and dynamics of singular quantum states generated by averaging of operator random walks

TL;DR

This paper investigates the behavior of quantum states under random channels, providing conditions for convergence and analyzing the evolution of states, especially focusing on singular states and their dynamics in quantum systems.

Contribution

It introduces new conditions for convergence of quantum state distributions under random unitary channels and analyzes the dynamics of singular quantum states via operator quadratic forms.

Findings

Established sufficient conditions for convergence in probability.

Described generalized convergence in distribution with respect to weak operator topology.

Analyzed the transmission and evolution of pure and normal states to singular states.

Abstract

In this paper the random channels and their compositions in the space of quantum states are studied. For compositions of i.i.d. random unitary channels, the limit behaviour of probability distributions is described. The sufficient condition for convergence in probability is obtained. The generalized convergence in distribution w.r.t. weak operator topology is obtained. The analysis of transmission of pure and normal states to the set of singular states is done. The dynamics of quantum states is described in terms of the evolution of the values of quadratic forms of operators from the algebra that implements the representation of canonical commutation relations.