Qubit Pauli twirl ($\mathcal{T}(\rho )=I/2$)

For any qubit operator $A$,Expand $A$ in the Pauli operator basis. The identity term is fixed, while every nonidentity coefficient has two positive and two negative contributions and cancels. Thus forgetting a uniformly random Pauli key sends every density operator to the maximally mixed state. This is the dimension-two Heisenberg-Weyl twirling channel, since replacing $Y$ by $XZ$ changes only an irrelevant phase in the conjugation.