Proposition: A declared all-or-nothing collection model

For the completion of Section sec:completion, suppose:

1. capture succeeds with message-independent probability ${\eta }_{\mathrm{cap}}$;

2. on successful capture, the coherent logical branch is collected without distortion with probability $\chi$;

3. every other complete branch is message-independent, including its surviving early state;

4. the corresponding flags are accessible.

Then

Proof. The complete output is a flagged mixture of an intact trusted-qubit branch with probability ${\eta }_{\mathrm{cap}}\chi p_L(q)$ and constant branches. Apply Proposition prop:flags.