Proposition: Distillation changes the original proxy

After this operation:

1. the pair $|\beta ⟩$ is transferred exactly to $XB$;

2. the original decoded partner register $F$ is in $|0⟩$;

3. the original proxy-pass probability is

   $$\mathrm{Pr}({\Pi }_{\beta })={p_0}^2;$$

   for a maximally entangled pair it is $d^{-2}$;

4. the message remains exactly recoverable from $E$;

5. any inaccessible outputs added by this operation remain independent of the message.

Proof. In the decoded coordinates, the operation maps

This proves items 1 and 2. The original $FB$ marginal becomes $|0⟩⟨0{|}_F\otimes {\tau }_B$. Therefore

Undoing the reapplied $U$ still exposes the untouched message in $D$, proving item 4. All other decoded factors depend only on the fixed pair preparation, proving item 5.