Higher rank matricial ranges and hybrid quantum error correction

We introduce and initiate the study of a family of higher rank matricial ranges, taking motivation from hybrid classical and quantum error correction coding theory and its operator algebra framework. In particular, for a noisy quantum channel, a hybrid quantum error correcting code exists if and onl...

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Bibliographic Details
Published in:Linear & multilinear algebra 2021-04, Vol.69 (5), p.827-839
Main Authors: Cao, Ningping, Kribs, David W., Li, Chi-Kwong, Nelson, Mike I., Poon, Yiu-Tung, Zeng, Bei
Format: Article
Language:English
Subjects:
Binary system
Codes
Error correcting codes
Error correction
Error correction & detection
Hilbert space
hybrid codes
joint higher rank matricial ranges
matricial range
Numerical range
operator algebra
Operators (mathematics)
quantum channel
quantum error correction
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Summary:We introduce and initiate the study of a family of higher rank matricial ranges, taking motivation from hybrid classical and quantum error correction coding theory and its operator algebra framework. In particular, for a noisy quantum channel, a hybrid quantum error correcting code exists if and only if a distinguished special case of the joint higher rank matricial range of the error operators of the channel is non-empty. We establish bounds on Hilbert space dimension in terms of properties of a tuple of operators that guarantee a matricial range is non-empty and hence additionally guarantee the existence of hybrid codes for a given quantum channel. We also discuss when hybrid codes can have advantages over quantum codes and present a number of examples.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2020.1748852