The fidelity of density operators in an operator-algebraic framework

Abstract

Josza’s definition of fidelity [R. Jozsa, J. Mod. Opt. 41(12), 2315–2323 (1994)] 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 ∈ A for which τ(ρ) = 1. The second setting 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 in the von Neumann algebra setting are novel in that we focus on the Schrödinger picture rather than the Heisenberg picture, and they also yield a new proof of a theorem of Molnár [Rep. Math. Phys. 48(3), 299–303 (2001)] on the structure of fidelity-preserving quantum channels on the trace-class operators.