A one-shot instrument ledger and the adaptive separation

This section concerns the specified Born execution interface. An unknown admissible finite-capacity strategy is not assumed to produce linear completely positive maps.

Let $\Phi =\{{\Phi }_x{\}}_{x\in X}$ be a finite instrument from $s$ input qubits to $s^{\prime }$ output qubits: each ${\Phi }_x$ is completely positive and ${\sum }_x{\Phi }_x$ is trace preserving. Discard zero outcome maps and write $X_{+}$ for the remaining labels. For a Kraus family,

Set

The finite-dimensional Choi, Kraus, and Stinespring conventions are those of Watrous (Chapter 2).