Finite carriers, complete outputs, and access

All Hilbert spaces in the mathematical results below are finite dimensional. Write $\mathcal{L}(H)$ for the linear operators on a Hilbert space $H$ and $\mathcal{D}(H)$ for its density operators. A channel is a completely positive trace-preserving linear map. Let

be the entire logical code whose identity channel is the recovery target. The symbol $Q$ denotes an arbitrary untouched external reference.