Maximal noiseless code rates for collective rotation channels on qudits

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.