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BY 4.0 license Open Access Published by De Gruyter Open Access July 15, 2019

Infinite dimensional generalizations of Choi’s Theorem

  • Shmuel Friedland EMAIL logo
From the journal Special Matrices

Abstract

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 μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ 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.

MSC 2010: 46N50; 81Q10; 94A40

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Received: 2019-03-08
Accepted: 2019-06-15
Published Online: 2019-07-15

© 2019 Shmuel Friedland, by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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