Entropy turnover is not a decoder transition

For a pure input in the flagged model, the radiation state has orthogonal occupied and vacuum blocks, so

Its entropy is maximal at $q_j=1/2$, where the optimal recovery error is $a_d/2$, not zero. Also $h_2(q)=h_2(1-q)$, although the respective recovery errors $q{a}_d$ and $(1-q)a_d$ differ.

More generally, Proposition prop:spectral-separation can be applied at successive times with the same prepared ancilla spectrum ${\sigma }_{K,t}$. It gives identical selected entropy histories while one logical channel returns the code and the other retains it. Producing a turnover in ${\sigma }_{K,t}$ does not repair the hidden-code model. This is the entropy/recovery separation motivating the supplied Hawk01-B comparison CheungHawk01B.

If radiation is cumulatively retained and never reinteracts, more access cannot worsen unrestricted recovery: whenever ${\mathcal{N}}_s=P_{t\to s}{\mathcal{N}}_t$ with $P_{t\to s}$ CPTP,

Indeed a decoder at $s$ can be composed with the restriction. This availability statement is not a streaming-memory or efficient execution theorem CheungEn02C.