Finite hybrid interfaces

A finite local interface has observable algebra

where $L$ is finite and every ${\mathcal{H}}_{\lambda }$ is finite dimensional. Its state space is represented by unnormalized positive blocks:

The blocks include classical outcomes, occupation flags, and controller records. Tensor products give multi-site interfaces.

A complete instrument is a finite family of CP maps $\{{\Phi }_k{\}}_k$ whose sum is trace preserving. Its recorded channel is

Failure outcomes are included. Classical discard sums blocks without conditional normalization. We use the standard finite-dimensional CP-map framework Choi,Watrous.