= Commutation of computational-basis dephasing and phase gates
{title2=$\mathcal D_c\circ\mathcal P_\varphi=\mathcal P_\varphi\circ\mathcal D_c$}
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> $P_\varphi=\operatorname{diag}(1,e^{i\varphi})$ maps the same entry to $e^{-i\varphi}\rho_{01}$. Since these scalar factors commute, their <quantum channels> commute on every <density matrix>. More directly, an interaction $|j\rangle|e\rangle\mapsto|j\rangle|e_j\rangle$ 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>.
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