Playing with Fidelities

TL;DR

This paper extends the concept of partial fidelities from finite-dimensional quantum states to von Neumann algebras, providing new formulas, estimates, and a broader axiomatic framework for understanding fidelity in quantum physics.

Contribution

It generalizes the notion of partial fidelities to the vN-algebraic setting and introduces a new axiomatic system based on relative majorization and complete positivity.

Findings

Formulas and estimates for partial fidelity in semifinite vN-algebras

A new axiomatic framework for fidelity based on relative majorization

Discussion of fidelity concepts in quantum physics

Abstract

The notion of partial fidelities as invented recently by A.Uhlmann for pairs of finite dimensional density matrices will be extended to the vN-algebraic context and is considered and thoroughly discussed in detail from a mathematical point of view. Especially, in the case of semifinite vN-algebras formulae and estimates for the partial fidelity between the functionals of a dense cone of inner derived normal positive linear forms are obtained. Also, some generalities on the notion of fidelity in quantum physics are collected in an appendix, and another system of mathematical axioms for fidelity over density operators, which is based on the concept of relative majorization and which is intimately related to the concept of complete positivity, is proposed.