Definition: Complete finite network

A complete finite network consists of:

1. a finite acyclic wiring diagram of system and classical ports;

2. a finite-dimensional quantum space, or a finite classical algebra, on each port;

3. a CPTP map at each operation vertex;

4. explicit ports for every classical control and reported outcome;

5. declared source preparations, terminal outputs, and inaccessible output systems.

A finite instrument $\{{\mathcal{I}}_o{\}}_o$ is represented by its recorded channel

where each selective map is CP and ${\sum }_o{\mathcal{I}}_o$ is trace preserving.

Adaptive control is a channel controlled by preceding classical record ports. A finite adaptive tree can consequently be represented as a finite network with classical registers and inactive branches explicitly completed.