Radon-Nikodym derivatives of quantum operations

TL;DR

This paper reviews the Radon-Nikodym theorem for completely positive maps, explores its applications to the structure of quantum operations, and discusses how it can be used for norm estimates in quantum information theory.

Contribution

It provides alternative formulations of the Radon-Nikodym theorem for CP maps and applies it to analyze quantum operations and their differences.

Findings

Characterization of CP maps via Radon-Nikodym derivatives

Analysis of order intervals of quantum operations

Derivation of norm estimates for differences of quantum operations

Abstract

Given a completely positive (CP) map $T$, there is a theorem of the Radon-Nikodym type [W.B. Arveson, Acta Math. {\bf 123}, 141 (1969); V.P. Belavkin and P. Staszewski, Rep. Math. Phys. {\bf 24}, 49 (1986)] that completely characterizes all CP maps $S$ such that $T-S$ is also a CP map. This theorem is reviewed, and several alternative formulations are given along the way. We then use the Radon-Nikodym formalism to study the structure of order intervals of quantum operations, as well as a certain one-to-one correspondence between CP maps and positive operators, already fruitfully exploited in many quantum information-theoretic treatments. We also comment on how the Radon-Nikodym theorem can be used to derive norm estimates for differences of CP maps in general, and of quantum operations in particular.