Purifier distillation and intervention dependence

A compatibility proposal must specify what happens after an exterior operation on the encoded partner. Computational difficulty alone is not an algebraic answer. Harlow–Hayden studies resource restrictions for particular distillation tasks HH; we impose no such restriction in the finite example.

Append a $d$-dimensional register $X$ in $|0{⟩}_X$. Perform:

1. undo $U_{FD}$;

2. apply ${\mathrm{SWAP}}_{FX}$;

3. reapply $U_{FD}$.

This is a specified exterior operation on $E$ and its ancilla.