Lemma: Entanglement fidelity is not dimension-free diamond control

Let $\Phi$ be a CPTP channel on a $d$-dimensional message, and let

If

then

Proof. The trace distance from a density operator to the pure Bell state is at most $\sqrt{1-f}$. Every pure message-reference input, with reference compressed to dimension $d$, is a filtered Bell vector $(A\otimes I)|{\Phi }_d⟩$, with ${\Vert A\Vert }_{\infty }^2\le d$. Applying this filter to the normalized Choi difference bounds its trace norm by $d$ times the Choi trace norm. Convexity handles mixed inputs. The characterization of the diamond norm on reference-extended states gives the result. Channel half-diamond distance is also at most one Watrous (Chapter 3).