Proposition: Accessible dimension bounds whole-code recovery

For a finite accessible output of dimension $r\le d$,

Exact recovery with constant residual rank $s$ additionally requires $r\ge ds$.

Proof. The Bell-input state on $Q\otimes R$ is a mixture of pure vectors of Schmidt rank at most $r$. Applying each decoder Kraus operator cannot increase that rank. A normalized vector of Schmidt rank at most $r$ has squared Bell overlap at most $r/d$, by Cauchy–Schwarz on its Schmidt coefficients. Convexity and the Bell-projector test prove (eq:dimension-lower). The exact statement is Theorem thm:exact.