Corollary: An informative record prevents universal recovery

Let $\{{\Psi }_a\}$ be an instrument. If at least one effect ${\Psi }_a^{\ast }(I)$ is not a scalar multiple of $I$, then:

1. its nonselective matter channel ${\sum }_a{\Psi }_a$ has no CPTP left inverse;

2. its complete classical-record–matter channel $\mathcal{I}(X)={⨁}_a{\Psi }_a(X)$ also has no CPTP left inverse.

Both assertions concern recovery of every input state.

Proof. For (i), use the dilation with environment basis $|a,\alpha ⟩$. Measuring its environment gives the instrument probabilities. A constant complement would make every such probability independent of the input, contradicting the nonscalar effect. Apply Theorem thm:decouple.

For (ii), suppose $\mathcal{R}\mathcal{I}=\mathrm{id}$. Restrict the recovery to each classical output block, obtaining a CPTP map ${\mathcal{R}}_a$. The completely positive maps ${\mathcal{R}}_a{\Psi }_a$ sum to the identity, so Lemma lem:identity-branches gives ${\mathcal{R}}_a{\Psi }_a=p_a\mathrm{id}$. Taking traces and using trace preservation of ${\mathcal{R}}_a$ implies ${\Psi }_a^{\ast }(I)=p_{a}I$, again a contradiction.