Proposition: Informative records exclude whole-input recovery

For a finite recorded instrument

if some effect ${\Psi }_w^{\ast }(I)$ is nonscalar, no channel recovers every input in the full quantum algebra from the complete record-plus-residual output.

Proof. Choose branch Kraus operators $K_{w,\nu }$. Kraus operators of $\mathcal{N}$ are $A_{w,\nu }=|w{⟩}_R\otimes K_{w,\nu }$. If a recovery channel has Kraus operators $B_j$, then $B_j{A}_{w,\nu }$ is a Kraus family for the identity. The identity channel has rank-one Choi matrix, so each such operator is a scalar multiple of $I$. Completeness of the $B_j$ implies

In particular, summing the diagonal terms at fixed $w$ makes ${\sum }_{\nu }{K_{w,\nu }}^{†}{K}_{w,\nu }$ scalar, a contradiction.