Infinite dimensional generalizations of Choi’s Theorem

Special Matrices

Friedland Shmuel¹ 

[1] Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607-7045, USA;

关键词: completely positive maps;    stinespring’s dilation theorem;    choi’s theorem;    trace class operators;    quantum channels;    quantum subchannels;    hellwig-kraus representation;    46n50;    81q10;    94a40;   

DOI  :  10.1515/spma-2019-0006

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown