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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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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
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description 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.
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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
title Higher rank matricial ranges and hybrid quantum error correction
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