Theorem: Entropy histories do not certify code preservation

There are conserving sequential evolutions $G,L$ with the following properties. For every prefix including the endpoint,

Consequently their full accessible radiation density operators agree for input $|0⟩⟨0|$.

Their scalar radiation entropy histories agree for every pure code input, although their radiation density operators need not agree for those other inputs. At the endpoint,

The nonpreserving accessible channel retains the input in an inaccessible $d$-dimensional memory; it does not destroy it globally.

Proof. The retaining evolution acts as identity on its variable memory and as the preserving sequence on its fixed dummy preparation. Tracing the memory gives (eq:retainingchannel). All maps remain isometries, and the variable code labels have equal energy, so the declared energy accounting is unchanged.

For the preserving sequence, Theorem thm:information shows that the radiation entropy depends on the input only through $H(\rho )$. Every pure code input has $H(\rho )=0$, so its pre-endpoint entropy is

The endpoint radiation is an isometric encoding and is pure for every pure input. Its entropy is therefore zero. The retaining sequence emits the fixed dummy history, giving exactly the same scalar values.

At the endpoint, the retaining radiation is constant and its complement contains the original input isometrically. A decoder on constant radiation can only prepare a fixed state. Lemma lem:depolarization gives optimal error $c_d$. The same lemma and a left inverse for the retained input give leakage distance $c_d$. The preserving endpoint result is Theorem thm:endpoint.