Lemma: Feasible upper witness

If a Hermitian matrix $Z$ and $\nu \ge 0$ satisfy

then $\frac{1}{2}{\Vert \Delta \Vert }_{\diamond }\le \nu /2$.

Proof. The first inequalities imply $Z⪰0$. For a pure referenced input $\psi =(A\otimes I){\Phi }_d$, $\mathrm{Tr}(A^{†}A)=d$, its Hermitian output difference $H$ obeys $-W⪯H⪯W$, where

Testing against the positive and negative spectral projections of $H$ gives

For an arbitrary operator $X$, the Hermitian block dilation

preserves its trace norm and the output trace norm under a Hermiticity-preserving map. Jordan decomposition reduces Hermitian inputs to states, and convexity reduces states to pure states. Thus the bound covers the diamond norm.