{c}

Averaging conjugation by all <Heisenberg-Weyl operators> completely depolarizes a system:
$$
\frac1{d^2}\sum_{k,m}W_{k,m}AW_{k,m}^\dagger=(\operatorname{Tr}A)\frac Id.
$$
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 $(\operatorname{Tr}_BA)\otimes I_B/d$. This identity turns <joint convexity of quantum relative entropy> into monotonicity under <partial trace>.


Heisenberg-Weyl twirling channel

{c}
{title2=$W_{k,m}=X^kZ^m$}

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


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 Incoming links: Heisenberg-Weyl operator