Theorem: No label recovery from identical accessible inputs

Let $x$ have prior distribution ${\pi }_x$. If every $x$ gives the same density on all degrees of freedom participating in the apparatus, and the apparatus channel and independent preparations do not depend on $x$, then all output record–spectator states are identical.

Any classical estimator based on those outputs has success probability at most ${\max }_x{\pi }_x$. For equal priors on $L$ labels the bound is $1/L$.

Proof. A fixed linear channel maps identical inputs to identical outputs. Every subsequent measurement or postprocessing therefore has an $x$-independent output law $q(z)$. For an estimator $g(z)$,

The same conclusion holds if the returned spectator is measured as part of the output.