Theorem: Exact retention and threshold on the cut

The stored code states are

The specified decoder has score

This is also optimal over encodings and binary decoders for this fixed composite storage channel.

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

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

Proof. Depolarization multiplies every Bloch component by $\eta$, and its factors compose multiplicatively. Born readout gives (eq:retentionscore).

In the proof of Theorem thm:qubitoptimal, storage multiplies the nonconstant score term by $r_T$. The same code attains the resulting upper bound. Solving $Q_{\mathrm{Q}}(T)>3/4$ gives (eq:retentionthreshold).