Maximal noiseless code rates for collective rotation channels on qudits

TL;DR

This paper analyzes the structure of noiseless subsystems in collective rotation channels for qudits, determining maximum code rates and subsystem dimensions, with explicit results for qutrits, advancing quantum error correction methods.

Contribution

It extends the analysis of noiseless subsystems to general qudits and provides explicit maximum dimensions for qutrits, improving quantum error correction strategies.

Findings

Maximum noiseless code rate approaches 1 for suitable subsystems.

Maximum dimension of noiseless subsystems can be computed via discrete optimization.

Explicit maximum dimension for qutrits is obtained through analysis and software.

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.