Proposition: Constant-qubit calibration

For a constant qubit channel $\mathcal{N}(X)=\mathrm{Tr}(X)\tau$,

Proof. Every decoded channel is constant, with some output $\sigma$. On a maximally entangled input its output is $I/2\otimes \sigma$, whose Bell overlap is $1/4$. Testing the Bell projector gives half-diamond error at least $3/4$.

Preparing $I/2$ attains the bound. Indeed, the completely depolarizing qubit channel equals the uniform mixture of the four Pauli conjugations. Subtracting the identity and using the triangle inequality gives half-diamond distance at most $3/4$.