### Abstract

Given a completely positive (CP) map T, there is a theorem of the Radon-Nikodym type [W. B. Arveson, Acta Math. 123, 141 (1969); V. P. Belavkin and P. Staszewski, Rep. Math. Phys. 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.

Original language | English (US) |
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Pages (from-to) | 5003-5020 |

Number of pages | 18 |

Journal | Journal of Mathematical Physics |

Volume | 44 |

Issue number | 11 |

DOIs | |

State | Published - Nov 1 2003 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Mathematical Physics*,

*44*(11), 5003-5020. https://doi.org/10.1063/1.1615697