Proposition: Replacement-channel incompatibility

If ${\mathcal{N}}_{\mathrm{rep}}$ is constant and $d_{\diamond }(\mathcal{N},{\mathcal{N}}_{\mathrm{rep}})\le b$, then

Proof. For every decoder,

The decoded replacement channel is constant. For a Bell input, every constant channel has target Bell overlap $d^{-2}$, so its error is at least $1-d^{-2}$. The maximally mixed replacement attains that bound, as follows from its Weyl-unitary twirl and the triangle inequality. Taking the decoder infimum proves the claim.