Lemma: Bell-overlap error bounds

For a channel $\Lambda :\mathcal{B}({\mathbb{C}}^d)\to \mathcal{B}({\mathbb{C}}^d)$, with normalized Bell projector $P_d$ and $F=\mathrm{Tr}(P_d{J}_{\Lambda })$,

Proof. Testing $P_d$ on the Bell input proves the lower bound.

For a state $\rho$ and a pure projector $P$,

One can prove this by decomposing $\rho$ into pure states, using the pure-state trace-distance formula, convexity, and concavity of the square root FuchsVanDeGraaf.

For a Hermiticity-preserving linear map $\Delta$,

Indeed, write unit referenced vectors as $u=(A\otimes I){\Phi }_d$ and $v=(B\otimes I){\Phi }_d$, with $\Vert A{\Vert }_{\mathrm{HS}}=\Vert B{\Vert }_{\mathrm{HS}}=\sqrt{d}$. The image of $|u⟩⟨v|$ is $(A\otimes I)J_{\Delta }(B^{†}\otimes I)$, whose trace norm is at most $d\Vert J_{\Delta }{\Vert }_1$. A singular-value decomposition extends the bound to every trace-norm-one input and arbitrary reference dimension. Apply it to $\Delta =\Lambda -\mathrm{id}$, divide by two, and use the pure-target state bound.