Lemma: Orthogonal unitary mixtures and constant-channel error

Suppose unitary channels ${\mathrm{Ad}}_{U_j}$ have 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 output difference 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 replacement state $\sigma$, the Bell output is $I_Q/d\otimes \sigma$. Its overlap with the target Bell projector is $d^{-2}$. That projector therefore gives the lower bound $1-d^{-2}$.

The $d^2$ Weyl unitaries $X^p{Z}^q$ have orthogonal Bell outputs, and their uniform twirl is ${\mathcal{T}}_{I_C/d}$. The preceding equality gives the matching upper bound. For an arbitrary channel, the decoder which ignores its input and prepares $I_C/d$ attains this same allowance.