Frozen data

Fix:

1. a message dimension $m\ge 2$ and pair dimension $d\ge 2$;

2. bases for ${\mathcal{H}}_M\simeq {\mathcal{H}}_D\simeq {\mathcal{H}}_K\simeq {\mathbb{C}}^m$ and ${\mathcal{H}}_A\simeq {\mathcal{H}}_F\simeq {\mathcal{H}}_B\simeq {\mathbb{C}}^d$;

3. positive weights $p_j$, ${\sum }_j{p}_j=1$, and

   $$|\beta {⟩}_{AB}={\sum _{j=0}}^{d-1}\sqrt{p_j}|j{⟩}_A|j{⟩}_B;$$

4. a fixed unitary $U$ on ${\mathcal{H}}_E={\mathcal{H}}_F\otimes {\mathcal{H}}_D$;

5. the accessible final output ${\mathcal{H}}_R={\mathcal{H}}_E\otimes {\mathcal{H}}_B$ and complete residual ${\mathcal{H}}_K$.

There are no unassigned auxiliary purifications.

The word “pair” denotes the prepared finite vector above. The labels $F,B,D$ are output names, not spacetime locations.

Let $J_0:{\mathcal{H}}_M\otimes {\mathcal{H}}_A\otimes {\mathcal{H}}_B\to {\mathcal{H}}_F\otimes {\mathcal{H}}_D\otimes {\mathcal{H}}_B$ be the fixed relabeling and permutation

and define the structural unitary

Set