Abstract

We study a specified even-parity excitation-qubit extraction isometry, its accessible channel, and its complete complement. Exact formulas distinguish occupation transfer, occupation-basis coherence, subsystem purity, reference correlations, and deterministic quantum recovery.

For extraction probability $\eta$, we evaluate the complementary pair-distance leakage as $1-{\eta }^2$ and prove its recovery lower bound without importing another model. We restate and prove the constant-complement criterion for exact recovery. A specified decoder has half-diamond error $1-\eta$; its flagged instrument exposes off-code events without postselection. We also determine the maximal decoded Bell fidelity, $(1+4\eta -{\eta }^2)/4$, and construct a balanced decoder. Writing $t=1-\eta$, these give the improved deterministic recovery bounds

The endpoint optima are $3/4$ and $0$; an exact intermediate half-diamond optimum is not asserted.

A separately declared population-preserving dephasing experiment retains its additional environment. At full transfer its optimal recovery error is $(1-\lambda )/2$, although its Bell-input accessible marginal is independent of $\lambda$. Complete dephasing reaches the entanglement-breaking recovery floor $1/2$ while preserving perfect occupation delivery. Normalized-Choi and matrix-unit estimates provide explicit referenced certification interfaces. An independent continuum argument separates the abstract full-transfer endpoint from finite-time attainability. These are finite-channel results, not a gravitational evaporation model or a Page-curve construction.