Fidelity of density operator in an operator-algebraic framework

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 $τ$. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements $ρ\in A$ for which $τ(ρ)=1$. The second of our two settings is more operator theoretic: by fixing a faithful normal semifinite trace $τ$ 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 Molnár on the structure of quantum channels on the trace-class operators that preserve fidelity.

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