Weighted Number Operators on Bernoulli Functionals and Quantum Exclusion Semigroups

TL;DR

This paper introduces weighted number operators on Bernoulli functionals using quantum Bernoulli noises, analyzes their spectral properties, and constructs quantum exclusion semigroups with applications in quantum probability.

Contribution

It defines and studies weighted number operators on Bernoulli functionals, including spectral decompositions and conditions for boundedness, and constructs related quantum exclusion semigroups.

Findings

Spectral decompositions of weighted number operators are established.

Necessary and sufficient conditions for boundedness are derived.

Quantum Markov semigroups related to these operators are constructed and analyzed.

Abstract

Quantum Bernoulli noises (QBN, for short) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation (CAR) in equal-time. In this paper, by using QBN, we first introduce a class of self-adjoint operators acting on Bernoulli functionals, which we call the weighted number operators. We then make clear spectral decompositions of these operators, and establish their commutation relations with the annihilation as well as the creation operators. We also obtain a necessary and sufficient condition for a weighted number operator to be bounded. Finally, as application of the above results, we construct a class of quantum Markov semigroups associated with the weighted number operators, which belong to the category of quantum exclusion semigroups. Some basic properties are shown of these quantum Markov semigroups, and examples are also given.