Finite channels and regional algebras

Let $X$ be a finite site set with finite-dimensional Hilbert spaces ${\mathcal{H}}_x$. Controller, token, permission, and record registers must be included among these sites or among the explicitly named ports of a more general network. Set

All channels are complete and trace preserving. Measurements retain every outcome, including rejection and failure branches.

For a channel $\mathcal{C}$, its adjoint is defined by

Trace preservation is equivalent to ${\mathcal{C}}^{\ast }(I)=I$.

The following finite comparison is the ceiling associated with the snapshot–channel–adjoint interface of En05-A CheungEn05A. We give its complete tensor-factor proof.