Spectra and Variance of Quantum Random Variables

TL;DR

This paper explores the spectral properties and variance of bounded quantum random variables, introducing operator-valued variance and new measures of quantum noise within quantum probability theory.

Contribution

It establishes the spectral characterization of quantum random variables and develops a novel operator-valued variance framework for quantum noise analysis.

Findings

Gelfand spectrum matches hypoconvex hull of essential range

Operator-valued variance formulation for quantum variables

New measures of quantum and inherent noise introduced

Abstract

We study essentially bounded quantum random variables and show that the Gelfand spectrum of such a quantum random variable coincides with the hypoconvex hull of its essential range. Moreover, a notion of operator-valued variance is introduced, leading to a formulation of the moment problem in the context of quantum probability spaces in terms of operator-theoretic properties involving semi-invariant subspaces and spectral theory. As an application of quantum variance, new measures of random and inherent quantum noise are introduced for measurements of quantum systems, modifying some recent ideas of Polterovich.