Thursday, June 6, 2013

1306.0981 (Chi-Kwong Li et al.)

Quantum error correction for collective rotation channels on qudits    [PDF]

Chi-Kwong Li, Mikio Nakahara, Yiu-Tung Poon, Nung-Sing Sze
We consider collective rotation channels of qudits, i.e., quantum channels with operators in the set $${\mathcal E}(m,n) = \{\underbrace{U\otimes \cdots \otimes U}_n: U \in {\mathrm{SU}}(m)\}$$. This is done by analyzing the algebra decomposition of the algebra ${\mathcal A}(m,n)$ generated by ${\mathcal E}(m,n)$. We summarize the results for the channels on qubits, and obtain the maximum dimension of the noiseless subsystem which can be used as the quantum error correction code for the channel. Then we extend the results to general $m$. In particular, it is shown that quantum error correction rate, i.e., the number of correctable qudits over the number of transmitted 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 for the case when $m = 3$ is determined by a combination of mathematical analysis and the symbolic software Mathematica. In addition to the study of quantum error correction, the results also have implications in Lie theory and representation theory concerning the algebra ${\mathcal A}(m,n)$
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