Lemma: Bell-subspace conversion

For every such channel,

In particular, $F_{\mathrm{sub}}(\Phi )=1$ implies $\Phi ={\mathrm{Tr}}_S$.

Proof. The Choi input and effect $M=P_2^{Q{C}_{\mathrm{out}}}\otimes I_R$ give the lower bound, because the target expectation is one.

Write $F=F_{\mathrm{sub}}(\Phi )$. For $F>0$, purify $J_{\Phi }$, project that purification with $M$, and normalize. The original and projected pure states have overlap $\sqrt{F}$. After tracing the purifying system, the projected state has the form

Trace preservation of $\Phi$ fixes the input-reference marginal, so $J_{\Phi }$ has $R$-marginal $I_R/s$. Contraction under partial trace therefore gives

The triangle inequality yields

The input dimension is $2s$. Applying (eq:choiconversion) proves the upper bound. For $F=0$, the capped upper bound is immediate. For $F=1$, the same reasoning gives identical Choi states and hence identical channels.