Proposition: Accessible-dimension lower bound

If $\dim A=m<d$, every decoder has

Proof. A state on $Q\otimes A$ has a pure-state decomposition with Schmidt rank at most $m$. Local Kraus maps on $A$ cannot increase that rank. Every decoded Bell-output state therefore has Schmidt number at most $m$. A unit vector of Schmidt rank at most $m$ has squared Bell overlap at most $m/d$, by Cauchy–Schwarz. Convexity extends this bound to the decoded state. The Bell-projector test proves the claim.