Lemma: Choi-coordinate control

Let $E_1,\dots ,E_{d^4}$ be a Hermitian Hilbert–Schmidt orthonormal basis of matrices on $C\otimes C$. Set

Then $|a_j(\Lambda )|\le 1$. For a Hermiticity-preserving $\Delta$,

Proof. Each $E_j$ has operator norm at most its Hilbert–Schmidt norm one, proving the scalar bound on a state.

For completeness,

To see this, write unit referenced vectors as $u=(A\otimes I){\Phi }_d$ and $v=(B\otimes I){\Phi }_d$, where $\mathrm{Tr}(A^{†}A)=\mathrm{Tr}(B^{†}B)=d$. The output of $|u⟩⟨v|$ is $(A\otimes I)J_{\Delta }(B^{†}\otimes I)$, with trace norm at most $d\Vert J_{\Delta }{\Vert }_1$. A singular-value decomposition extends the inequality to all trace-norm-one inputs and all reference dimensions.

The matrix $J_{\Delta }$ has size $d^2$, so

Divide by two.