Theorem: Quantitative commuting-copy obstruction

Let $J:{\mathcal{H}}_L\to {\mathcal{H}}_{\mathrm{phys}}$ be an isometry, and let $X,Z\in \mathcal{B}({\mathcal{H}}_L)$ be contractions. Let ${\hat{X}}_1$ and ${\hat{Z}}_2$ also be contractions, satisfying

Then

In particular, if the physical representatives commute and $X,Z$ are logical Pauli operators on a qubit factor, then

Two exactly intertwining commuting representations cannot both represent the same full noncommutative logical factor.

Proof. First,

Each line has norm at most ${\epsilon }_1+{\epsilon }_2$. Subtract, apply the triangle inequality, and use ${\Vert J[X,Z]\Vert }_{\infty }={\Vert [X,Z]\Vert }_{\infty }$. For logical Pauli operators, ${\Vert [X,Z]\Vert }_{\infty }=2$.