Finite states and destructive readout

For a finite labelled register $\Gamma$, let ${\mathcal{H}}_{\Gamma }$ be its finite-dimensional Hilbert space. Write $L({\mathcal{H}}_{\Gamma })$ for its operators and $\mathcal{D}({\mathcal{H}}_{\Gamma })$ for its density operators. A channel is a completely positive, trace-preserving linear map.

Quantum names are access rights to factors of one joint state. Classical registers have distinguished bases. Admissible states are jointly block diagonal in those bases, including their correlations with all other registers. A diagonal classical marginal alone is not the required condition.

Born destructive computational-basis readout is a specified primitive. For a measured factor $S$, complement $B$, and classical output record $R$, put

Its channel is

The factor $S$ is absent from the output. No uniqueness theorem for the Born law is needed as a premise.

Coherent recording instead uses the isometry

where $A$ is a fresh quantum carrier. It retains both $S$ and $A$. A nondestructive instrument is another interface and must be specified separately.