Heisenberg-Weyl operator ($W_{k,m}=X^k{Z}^m$)

On a $d$-dimensional Hilbert space, let $X|j⟩=|j+1\mathrm{mod}d⟩$ and $Z|j⟩=e^{2\pi ij/d}|j⟩$. The $d^2$ unitaries $W_{k,m}=X^k{Z}^m$ form an orthogonal operator basis under the Hilbert-Schmidt inner product.

Averaging conjugation by all Heisenberg-Weyl operators completely depolarizes a system:Summing over phases $Z^m$ kills off-diagonal matrix elements, and averaging shifts $X^k$ makes all diagonal elements equal. Acting on subsystem $B$ of a bipartite operator gives $({\mathrm{Tr}}_{B}A)\otimes I_B/d$. This identity turns joint convexity of quantum relative entropy into monotonicity under partial trace.

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.