Theorem: Dynamical recovery with the complete residual retained

For the Hamiltonian, preparation, code, and access prescription of Section sec:model,

When $p_M(t)>0$, a complete decoder attaining the recovery upper bound is

Moreover,

No small-leakage hypothesis is imposed: the right-hand side is calculated from the autonomous Hamiltonian.

Proof. Suppose first that $s=s_M(t)<1$. The ideal complete code isometry is

For each label, the one-particle overlap between its actual evolved vector and its ideal collected vector is $\sqrt{1-s}$. Second quantization therefore gives, on source Fock space,

where the collected modes are included in the full output space and $N_S$ denotes input source particle number. Consequently,

This operator is positive and self-adjoint. Hence

The first term of (eq:complete-decoder) is completely positive. The second is a positive-effect measurement followed by state preparation, and is completely positive. Their traces add to $\mathrm{Tr}Y$, so the decoder is CPTP. It exactly decodes $V_{0,t,M}$.

Lemma lem:isometry and postprocessing contraction give the recovery bound. The ideal complementary channel is ${\mathcal{T}}_{|\mathrm{vac}⟩⟨\mathrm{vac}|}$, so the same lemma gives the complete residual bound.

All half-diamond distances between channels are at most one. If $s=1$, use this trivial bound and any complete decoder.

Finally,

where the second inequality is Bernoulli's inequality for the integer $K$. This proves (eq:simple-beta).