Theorem: Residual obstruction and its robust form

Record compatibility does not determine a residual instrument. There are ideal targets with identical records on every input whose outputs have trace distance $1/2$ on one qubit input.

If implemented versions have errors at most ${ϵ}_{\mathcal{F}},{ϵ}_{\mathcal{G}}$ on that input, their mutual distance is at least

Proof. Take $H=K={\mathbb{C}}^2$, $T=\mathrm{id}$, the computational projectors, and one coarse outcome $\ast$ with ${\mathcal{I}}_{\ast }=\mathrm{id}$. Both record effects are $I$. For ${\rho }_{+}=|+⟩⟨+|$,

Their difference has residual block

with eigenvalues $1/2,-1/2$. Its trace norm is one. The robust bound is the triangle inequality.