## Radon-Nikodym derivatives of quantum operations (2003)

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Venue: | JOURNAL OF MATHEMATICAL PHYSICS |

Citations: | 3 - 2 self |

### BibTeX

@ARTICLE{Raginsky03radon-nikodymderivatives,

author = {Maxim Raginsky},

title = {Radon-Nikodym derivatives of quantum operations},

journal = {JOURNAL OF MATHEMATICAL PHYSICS},

year = {2003},

volume = {44},

pages = {5003--5020}

}

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### 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.

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Citation Context .... However, it is not clear how to apply this theorem directly to CP maps given in terms of a “continual” Kraus decomposition (as in, e.g., the quantum operational model of Gaussian displacement noise =-=[12]-=-). For example, if Ug is a strongly continuous unitary representation of a compact topological group G on a Hilbert space H , how do we describe all CP maps completely dominated by the channel ∫ T(A) ... |

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Radon-Nikodym derivatives of quantum instruments
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Citation Context ...ring of positive measures or positive linear functionals. There are a number of Radon-Nikodym theorems for CP maps (see, e.g., the work of Arveson [1], Belavkin and Staszewski [2], Davies [9], Holevo =-=[14]-=-, Ozawa [24], and Parthasarathy [26]) that differ widely in scope and in generality. Thus, the results of Davies, Ozawa, and Holevo have to do with RadonNikodym derivatives of CP instruments [24] with... |

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Citation Context ...mount importance to have at one’s disposal a good analysis toolkit for completely positive (CP) maps. There are many useful structure theorems for CP maps. The two best known ones, due to Stinespring =-=[34]-=- and Kraus [20], are de rigueur in virtually all quantum information-theoretic treatments. These theorems are significant because each of them states that a given map is CP if and only if it is expres... |

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Citation Context ...maps). More recently, thisRadon-Nikodym Derivatives of Quantum Operations 13 correspondence has been exploited fruitfully in some quantum information-theoretic contexts, such as optimal cloning maps =-=[7]-=-, optimal teleportation protocols [16], separability criteria for entangled states [22], or entanglement generation [6, 37]. In this section we will show that the one-to-one correspondence between pos... |

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Citation Context ...Raginsky where TrE (·) denotes the partial trace over E , ξ ∈ E is a fixed unit vector, and U is the unitary extension of the partial isometry Û from H2 ⊗ [|ξ〉〈ξ|] to H1 ⊗ E defined by Û(ψ ⊗ ξ) = V ψ =-=[20, 23]-=-. (We use [P] to denote the closed subspace corresponding to the orthogonal projection P.) Finally, note that the input and output Hilbert spaces do not have to be the same; in general, quantum operat... |

1 |
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Citation Context ...e linear functionals. There are a number of Radon-Nikodym theorems for CP maps (see, e.g., the work of Arveson [1], Belavkin and Staszewski [2], Davies [9], Holevo [14], Ozawa [24], and Parthasarathy =-=[26]-=-) that differ widely in scope and in generality. Thus, the results of Davies, Ozawa, and Holevo have to do with RadonNikodym derivatives of CP instruments [24] with respect to scalar measures. On the ... |