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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 |
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creator | Cao, Ningping Kribs, David W. Li, Chi-Kwong Nelson, Mike I. Poon, Yiu-Tung Zeng, Bei |
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. |
doi_str_mv | 10.1080/03081087.2020.1748852 |
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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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