Bistochastic operators and quantum random variables

TL;DR

This paper develops a framework for quantum majorization using bistochastic operators on quantum random variables, extending classical majorization theory to the quantum operator setting with considerations for continuity and convergence.

Contribution

It introduces a new Banach space of quantum random variables and defines quantum majorization via bistochastic operators, bridging classical and quantum majorization theories.

Findings

Defined a seminorm leading to a Banach space of quantum random variables

Established a quantum majorization concept using bistochastic operators

Connected quantum majorization to inequalities involving convex functions

Abstract

Given a positive operator-valued measure $\nu$ acting on the Borel sets of a locally compact Hausdorff space $X$, with outcomes in the algebra $\mathcal B(\mathcal H)$ of all bounded operators on a (possibly infinite-dimensional) Hilbert space $\mathcal H$, one can consider $\nu$-integrable functions $X\rightarrow \mathcal B(\mathcal H)$ that are positive quantum random variables. We define a seminorm on the span of such functions which in the quotient leads to a Banach space. We consider bistochastic operators acting on this space and majorization of quantum random variables is then defined with respect to these operators. As in classical majorization theory, we relate majorization in this context to an inequality involving all possible convex functions of a certain type. Unlike the classical setting, continuity and convergence issues arise throughout the work.