Infinite dimensional generalizations of Choi’s Theorem

In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for...

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Bibliographic Details
Published in:Special matrices 2019-01, Vol.7 (1), p.67-77
Main Author: Friedland, Shmuel
Format: Article
Language:English
Subjects:
46N50
81Q10
94A40
Choi’s theorem
Completely positive maps
Hellwig-Kraus representation
quantum channels
quantum subchannels
Stinespring’s dilation theorem
trace class operators
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Summary:In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map is called a quantum channel, if it is trace preserving, and is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for to be a quantum subchannel.We show that is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.
ISSN:2300-7451
2300-7451
DOI:10.1515/spma-2019-0006