Proposition: SWAP-route commutator identity

Let $V_t$ be a finite SWAP permutation of qubit sites, with permutation ${\pi }_t$, and let $D_{\mathrm{all}}$ be the total Hilbert-space dimension. Then

Proof. The Heisenberg observable is $Z_{{{\pi }_t}^{-1}(y)}$. It commutes with $X_x$ on distinct sites. On the same site, the commutator's squared modulus is $4I$.