Definition: Admissible finite realization

At refinement $j$, a finite realization consists of:

1. a finite set $X\subset {\Lambda }_j$ containing a source $0$ and the declared destination;

2. one register $A_v\cong {\mathbb{C}}^D$ at each $v\in X$;

3. a prescribed finite sequence of partitions ${\mathcal{B}}_1,\dots ,{\mathcal{B}}_n$ of $X$;

4. for each tick $k$, a channel

   $$C_k=\underset{B\in {\mathcal{B}}_k}{⨂}{C}_{k,B}$$

   on $\mathcal{L}({\mathcal{H}}_X)$, where

   $${\mathcal{H}}_X=\underset{v\in X}{⨂}{A}_v;$$

5. blocks $B$ of cardinality at most two, with $\mathrm{diam}(B)\le a_j$, and completely positive trace-preserving maps $C_{k,B}$ on the fixed block registers.

Each tick is exactly one such partition layer and takes time ${\delta }_j$. The partitions and channels are fixed before the payload is supplied and do not depend on its state.