Radon-Nikodym derivatives of quantum operations

Details Radon-Nikodym derivatives of quantum operations Radon-Nikodym derivatives of quantum operations

2003-03-25

arxiv.org/abs/math-ph/0303056v3

J. Math. Phys. 44, 5003-5020 (2003)

Physics

Mathematical Physics

22 pages; final version

Scientific paper

10.1063/1.1615697

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.

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