Proposition: One-shot environment accounting

For a finite instrument $\{{\mathcal{A}}_x{\}}_x$, let $r_x=\mathrm{rank}J({\mathcal{A}}_x)$ after zero maps are removed, and $R_0={\sum }_x{r}_x$. A pure-environment isometric implementation with orthogonal outcome blocks has minimum environment dimension $R_0$. Its full-readout environment register needs

qubits. A prescribed rectangular outcome/index split needs

Proof. The minimum Kraus number of a CP map is its Choi rank: spectral decomposition supplies that many Kraus vectors, while any Kraus decomposition is a positive sum of rank-one Choi matrices and cannot use fewer. Orthogonal outcome blocks add their dimensions. The displayed encodings attain the bounds.