Fidelity of density operator in an operator-algebraic framework

TL;DR

This paper investigates the concept of fidelity for quantum states within operator algebras, establishing its properties and invariance under certain transformations, and provides a new proof related to quantum channels.

Contribution

It extends the definition and analysis of quantum fidelity to algebraic and operator-theoretic frameworks, including new proofs of fidelity-preserving properties of quantum channels.

Findings

Fidelity is monotone under specific trace-preserving positive maps.

Fidelity is preserved by certain quantum channels.

Provides a new proof of Molnár's theorem on fidelity-preserving quantum channels.

Abstract

Josza's definition of fidelity for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C$^*$-algebras $A$ that possess a faithful trace functional $\tau$. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements $\rho\in A$ for which $\tau(\rho)=1$. The second of our two settings is more operator theoretic: by fixing a faithful normal semifinite trace $\tau$ on a semifinite von Neumann algebra $M$, we define and consider the fidelity of pairs of positive operators in $M$ of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of $A$ or of the predual $M_*$. Our results also yield a new proof of a theorem of Moln\'ar on the structure of quantum channels on the trace-class operators that preserve fidelity.