Lemma: Choi and complete channel distances

For channels with input dimension $n$,

Proof. The normalized Bell input gives the lower bound.

For the upper bound, purify any referenced input. By its Schmidt decomposition it can be written, up to an isometry on the reference, as

The output difference is the corresponding multiplication of $J(\Phi )-J(\Psi )$ by $A$ and $A^{†}$. The trace-norm ideal property bounds its half trace norm by $n\delta$. Partial trace of a purification cannot increase trace distance. The distance between channels is at most one.