Lemma: Choi and pure-target bounds

For channels with common input dimension $d$,

For a state $\rho$, a rank-one projector $P$, and $F=\mathrm{Tr}(P\rho )$,

Consequently,

Proof. A Bell input proves the first lower bound. For $\Delta =\Phi -\Psi$, write unit referenced vectors as

Then

A singular-value decomposition extends this to all trace-norm-one inputs and all reference dimensions. Divide by two and use the channel-distance cap one.

The effect $P$ proves the pure-target lower bound. Decompose $\rho ={\sum }_k{p}_k|{\psi }_k⟩⟨{\psi }_k|$. Pure-state trace distance is $\sqrt{1-|⟨\varphi ,{\psi }_k⟩{|}^2}$ for $P=|\varphi ⟩⟨\varphi |$. Trace-norm convexity and square-root concavity give the upper bound. Combining the two inequalities proves (eq:bellbounds).