Proposition: No commuting duplicate action

For $d\ge 2$, there cannot be two commuting star representations ${\pi }_1,{\pi }_2$ of $\mathcal{B}(L)$ on the joint output which preserve the code and satisfy

for all $X\in \mathcal{B}(L)$.

Proof. For arbitrary $X,Y$,

Injectivity of $V$ would make every logical matrix commute. This is false for $d\ge 2$.