$\mu$-norm and regularity

TL;DR

This paper revisits the $40-norm for bounded operators in Hilbert spaces, extending its properties and exploring its role in constructing quantum entropy through bistochastic operators.

Contribution

It refines the understanding of the $40-norm, identifying classes of unitary operators that produce bistochastic operators for quantum entropy applications.

Findings

Identifies three classes of unitary operators generating bistochastic operators.

Provides further results on the properties of the $40-norm.

Lays groundwork for using the $40-norm in quantum entropy construction.

Abstract

In \cite{Tre_PSI20} we introduce the concept of a $\mu$-norm for a bounded operator in a Hilbert space. The main motivation is the extension of the measure entropy to the case of quantum systems. In this paper we recall the basic results from \cite{Tre_PSI20} and present further results on the $\mu$-norm. More precisely, we specify three classes of unitary operators for which the $\mu$-norm generates a bistochastic operator. We plan to use the latter in the construction of quantum entropy.