Commutation of computational-basis dephasing and phase gates (${\mathcal{D}}_c\circ {\mathcal{P}}_{\phi }={\mathcal{P}}_{\phi }\circ {\mathcal{D}}_c$)

For a qubit, computational basis dephasing with complex overlap $c$ maps ${\rho }_{01}$ to $\overline{c}{\rho }_{01}$ and preserves the diagonal entries. Conjugation by the phase gate $P_{\phi }=\mathrm{diag}(1,e^{i\phi })$ maps the same entry to $e^{-i\phi }{\rho }_{01}$. Since these scalar factors commute, their quantum channels commute on every density matrix. More directly, an interaction $|j⟩|e⟩\mapsto |j⟩|e_j⟩$ and a diagonal phase gate give the same joint quantum state in either order on the specified initial environment state. This need not hold for an interaction that changes the quantum state populations.