Two-body control of noiseless subsystems with mixed $SU(3)$ representations lifts teleportation above the classical fidelity limit
Collective noise can leave degrees of freedom unaffected, allowing them to store quantum information. Processing this information also requires physically available logical gates. We consider $r$ fundamental qutrits transforming under $U\in SU(3)$ and $s$ dual qutrits transforming under its complex conjugate $\overline U$, with collective action $R_{r,s}(U)=U^{\otimes r}\otimes\overline U^{\otimes s}$. Using the image of the walled Brauer algebra and Lie-algebra arguments, we show that adjacent exchanges within each qutrit type and one singlet-projector interaction between the two types generate any independently chosen family of special unitary gates on the noiseless subsystems by a finite pulse sequence, for every $r,s\geq1$. This establishes semi-universality of this system. We illustrate the result with a numerically optimised sequence of 24 pulses for three fundamental qutrits and one dual qutrit. We then use the encoding and available logical corrections for qutrit teleportation under correlated Weyl noise. Under explicit assumptions on channel composition and which trials are retained, and allowing for baseline implementation errors, we identify noise ranges in which encoded teleportation exceeds the classical fidelity bound of $1/2$, while the unencoded teleportation channel is entanglement breaking. We analyse fidelity on accepted trials separately from the transmission costs of the additional physical qutrits.