Maximal noiseless code rates for collective rotation channels on qudits

被引：0
作者：
Chi-Kwong Li
Mikio Nakahara
Yiu-Tung Poon
Nung-Sing Sze
机构：
[1] College of William & Mary,Department of Mathematics
[2] Kinki University,Research Center for Quantum Computing, Graduate School of Science and Engineering
[3] Kinki University,Department of Physics
[4] Shanghai University,Department of Mathematics, and Department of Physics
[5] Iowa State University,Department of Mathematics
[6] The Hong Kong Polytechnic University,Department of Applied Mathematics
来源：
Quantum Information Processing | 2015年 / 14卷
关键词：
Quantum error correction; Special unitary groups; Irreducible representations; Qudits;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
We study noiseless subsystems on collective rotation channels of qudits, i.e., quantum channels with operators in the set E(d,n)={U⊗n:U∈SU(d)}.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}(d,n) = \{ U^{\otimes n}: U \in {\mathrm {SU}}(d)\}.$$\end{document} This is done by analyzing the decomposition of the algebra A(d,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}(d,n)$$\end{document} generated by E(d,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}(d,n)$$\end{document}. We summarize the results for the channels on qubits (d=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2$$\end{document}) and obtain the maximum dimension of the noiseless subsystem that can be used as the quantum error correction code for the channel. Then we extend our results to general d. In particular, it is shown that the code rate, i.e., the number of protected qudits over the number of physical qudits, always approaches 1 for a suitable noiseless subsystem. Moreover, one can determine the maximum dimension of the noiseless subsystem by solving a non-trivial discrete optimization problem. The maximum dimension of the noiseless subsystem for d=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d = 3$$\end{document} (qutrits) is explicitly determined by a combination of mathematical analysis and the symbolic software Mathematica.
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页码：4039 / 4055
页数：16
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