Proposition: Physical logical twirling defines a Pauli task

For any qubit channel $\Lambda$, the channel

is Pauli and has $F({\Lambda }^{\mathrm{tw}})=F(\Lambda )$. It can be implemented by choosing an independent uniform Pauli before encoding, applying its inverse after complete decoding, and forgetting the random choice.

Proof. Choose a Kraus family and expand each Kraus operator in the Pauli basis. The resulting positive process matrix has trace one. Conjugation by a Pauli multiplies each Pauli component by its commutation sign. Averaging over the four signs cancels every off-diagonal process entry and retains the nonnegative diagonal entries. Thus the average is a Pauli channel. The identity process entry is unchanged, and it equals Bell overlap. The described random program is exactly the displayed convex mixture.