Proposition: A register-level realization

Let $C$ consist of $n$ qubits. Prepare a blank $n$-qubit register $M$ and swap the code into it, leaving the old visible register blank. For each $j$, swap $M_j$ into a fresh blank outgoing qubit $S_j$. Before outgoing readout, the first $k$ emitted qubits together with the untransferred part of $M$ contain the entire input, up to fixed register permutations and blank factors.

If each $S_j$ is destructively read immediately, the record after $k$ emissions is

An added destructive readout-and-drop of the remaining quantum core does not change (eq:toy-record).

Proof. Every SWAP is unitary and only relocates a tensor factor. After $k$ transfers, the logical tensor factors have been permuted into $S_1,\dots ,S_k,M_{k+1},\dots ,M_n$. All transferred positions of $M$ are blank. This proves complete preservation, including on entangled inputs.

The computational measurement effects on the transferred factors pull back to $|x⟩⟨x|\otimes I$ on the original code. This gives (eq:toy-record). The terminal operation acts only on the remaining quantum factors and returns none of its outputs, so Theorem thm:drop leaves the earlier record unchanged.