Lemma: Orthogonal unitary mixtures and constant reconstruction

Suppose the unitary channels ${\mathrm{Ad}}_{U_j}$ have mutually orthogonal Bell outputs. For real coefficients $a_j$,

In particular,

Every channel from $C$ has recovery error at most $a_d$.

Proof. A Bell input gives a diagonal signed output with eigenvalues $a_j$, proving the lower bound. Each unitary channel has diamond norm one, so the triangle inequality gives the upper bound.

For any state $\sigma$, the Bell output of ${\mathcal{T}}_{\sigma }$ is $I_Q/d\otimes \sigma$. Its overlap with the target Bell projector is $d^{-2}$. Testing that projector therefore gives

The $d^2$ Weyl unitaries have orthogonal Bell outputs and uniform twirl ${\mathcal{T}}_{{\tau }_C}$. The first part gives the matching upper bound.

A decoder that ignores its input and prepares ${\tau }_C$ attains the same error after any channel.