Fidelity of density operator in an operator-algebraic framework

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${}^{\ast }$-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 $M_{\ast }$. 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.

To appear in Journal of Mathematical Physics

References in corpus (4)

• Completely positive maps of order zero
• Monotonicity of the Quantum Relative Entropy Under Positive Maps
• Some Formulae for Norms of Elementary Operators
• Private algebras in quantum information and infinite-dimensional complementarity

Cited by in corpus (1)

• Multiplicative Properties of Quantum Channels