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arXiv · 1306.0981

Maximal noiseless code rates for collective rotation channels on qudits

Chi-Kwong Li·Mikio Nakahara·Yiu-Tung Poon·Nung-Sing Sze
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Abstract

We study noiseless subsystems on collective rotation channels of qudits, i.e., quantum channels with operators in the set ${\mathcal E}(d,n) = \{ U^{\otimes n}: U \in {\mathrm{SU}}(d)\}.$ This is done by analyzing the decomposition of the algebra ${\mathcal A}(d,n)$ generated by ${\mathcal E}(d,n)$. We summarize the results for the channels on qubits ($d=2$), 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 $d = 3$ (qutrits) is explicitly determined by a combination of mathematical analysis and the symbolic software Mathematica.

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Chi-Kwong Li, Mikio Nakahara, Yiu-Tung Poon, Nung-Sing Sze. 2013-06-05. Maximal noiseless code rates for collective rotation channels on qudits. https://arxiv.org/abs/1306.0981

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