ar X iv : 1 60 2 . 08 17 7 v 1 [ qu an t - ph ] 2 6 Fe b 20 16 The Fidelity of Density Operators in an Operator Algebraic Framework ∗

Josza’s definition of fidelity [10] 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 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 [15] on the structure of quantum channels on the trace-class operators that preserve fidelity. Introduction In communication and information theory, the notion of fidelity provides a quantitative measure for the qualitative assessment of how well data or information has been preserved through some type of transmission procedure or information processing task. Not surprisingly, this concept appears in quantum information theory [3, 10, 22] as well, with the aim of providing a similar quantitative measure. In a rather different form (namely, in the guise of the Bures metric [2, 4, 12]), the notion of fidelity also occurs in operator algebra theory. In one of the most important settings, fidelity is a numerical measure of how close one state σ of a quantum system is to another state ρ. For pure states— that is, for unit vectors ξ and η in a separable Hilbert space H—the fidelity is 2010 Mathematics Subject Classification: Primary 46L05; Secondary 46L60, 81R15