Example: A constant-error guarantee permits persistent error

The qubit channel

has $d_{\diamond }({\mathcal{Z}}_{ϵ},\mathrm{id})=ϵ$: the Bell outputs give the lower bound, and the triangle inequality gives the upper bound. For $ϵ=1/4$, it obeys the documentary tolerance, but repeated use has error

A constant upper allowance is not an actual lower bound on every implementation, but it supplies no vanishing-error conclusion.