Recursive encoding and decoding of the noiseless subsystem for qudits

Abstract

We give a full explanation of the noiseless subsystem that protects a single qubit against collective errors and the corresponding recursive scheme described by C.-K. Li et al. [Phys. Rev. A 84, 044301 (2011)] from a representation theory point of view. Furthermore, we extend the construction to qudits under the influence of collective SU(𝑑) errors. We find that under this recursive scheme, the asymptotic encoding rate is 1/𝑑.