Theorem: Exact depolarizing retention and optimal score

The stored code states are

The projective decoder gives

This is also the optimum over all encodings and binary decoders for the specified input-independent depolarizing storage channel.

Its strict violation of the perfect classical-storage benchmark occurs exactly when

For constant $\eta$, $r_T={\eta }^T$.

Proof. Replacement and identity compose to another depolarizing channel, with multiplicative Bloch factor. Apply Born's rule to the displayed states.

For arbitrary input Bloch vectors, the proof of Theorem thm:quantumoptimal acquires a common multiplier $r_T$ on its nonconstant term. The same square encoding attains that upper bound. Finally solve