Entropy of a unitary operator on $L^2(\pmb{\mathbb{T}}^n)$

Abstract

The concept of the $\mu$-norm of an operator, introduced in [28], is investigated. The focus is on operators on $L^2(\mathbb{T}^n)$, where $\mathbb{T}^n$ is the $n$-torus (the case when $n=1$ was previously considered in [29]). The main source of motivation for the study was the role of the $\mu$-norm as a key tool in constructing a quantum analogue of metric entropy, namely, the entropy of a unitary operator on $L^2(\mathcal X,\mu)$, where $(\mathcal X,\mu)$ is a probability space. The properties of the $\mu$-norm are presented and some ways to calculate it for various classes of operators on $L^2(\mathbb{T}^n)$ are described. The construction of entropy proposed in [28] is modified to make it subadditive and monotone with respect to partitions of $\mathcal X$. Examples of the calculation of entropy are presented for some classes of operators on $L^2(\mathbb{T}^n)$. Bibliography: 29 titles.