Proposition: Classical destructive access is not coherent recovery

Suppose the only accessible output of an input system is a finite classical measurement record. Any channel obtained by decoding that record into a quantum output has the form

and is entanglement breaking. It cannot equal the identity channel on a code of dimension at least two.

Proof. With a reference, the output is a sum of positive reference conditional operators tensored with ${\sigma }_z$, hence is separable. The identity channel preserves a maximally entangled state on a code of dimension at least two, whose output is not separable.