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arXiv:1706.07945

On the origin of non-decomposable maps

W. A. Majewski

Published Jun 24, 2017• 0 citations
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Abstract

The Radon-Nikodym formalism is used to study the structure of the set of positive maps from $\mathcal{B}(\mathcal{H})$ into itself, where $\mathcal{H}$ is a finite dimensional Hilbert space. In particular, this formalism was employed to formulate simple criteria which ensure that certain maps are non decomposable. In that way, a recipe for construction of non decomposable maps was obtained.

DOI
10.48550/arxiv.1706.07945

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The Radon-Nikodym formalism is used to study the structure of the set of positive maps from B(H)\mathcal{B}(\mathcal{H})B(H) into itself, where H\mathcal{H}H is a finite dimensional Hilbert space. In particular, this formalism was employed to formulate simple criteria which ensure that certain maps are non decomposable. In that way, a recipe for construction of non decomposable maps was obtained.
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.
A well-known theorem of W. Arveson states that a completely positive (CP) map dominated (difference is CP) by a given CP map is described through a positive contraction in the commutant of the homomorphism of Stinespring representation. This contraction plays the role of Radon-Nikodym derivative of measure theory. We look at implications of this to one parameter semigroups of unital completely positive maps, known as quantum Markov semigroups. We focus our attention to quantum Markov semigroups of B(H), the algebra of all bounded operators on a Hilbert space. These semigroups can be dilated to semigroups of unital endomorphisms (E₀-semigroups) of B(K) for some other Hilbert space K. Here we see that dominated CP semigroups dilate to dominated semigroups of the dilation. Moreover at the level of Eo-semigroups the dominated CP semigroups are described through positive contractive local cocycles, which are really families of contractions of 'Radon-Nikodym derivatives' mentioned above. We analyze the structure for quantum semigroups with bounded generators.