Lemma: Feasible diamond upper witness

Let $\Delta$ be Hermiticity preserving and $K_{\Delta }=d{J}_{\Delta }$. If a Hermitian $Z$ and $\nu \ge 0$ satisfy

then

Proof. For a pure referenced input $\psi =(A\otimes I){\Phi }_d$, $\mathrm{Tr}(A^{†}A)=d$, its Hermitian output difference $H$ satisfies $-W⪯H⪯W$, where

The first two matrix inequalities imply $Z⪰0$. Testing against the positive and negative spectral projections of $H$ gives

For arbitrary input operators, Hermitian block dilation preserves input and output trace norms for a Hermiticity-preserving map. Jordan decomposition reduces Hermitian inputs to states; convexity reduces states to pure states. This proves the diamond bound.