Completely positive maps can be characterized without involving external systems by means of the Stinespring factorization theorem, which reduces to Choi's theorem for the case of finite dimensional Hilbert spaces:

$ \Phi(a) = \sum_{i=1}^{mn}V_i^\dagger a V_i$.

Completely positive maps are considered to represent the most general quantum evolutions, however, Shaji and Sudarshan:

"ftp://79.110.128.93/books/physics,%20math/%D0%96%D1%83%D1%80%D0%BD%D0%B0%D0%BB%D1%8B/Phys%20Letters%20A/Volume%20341,%20Issues%201-4,%20pp.%201-356%20(20%20June%202005)/Anil%20Shaji,%20E.C.G.%20Sudarshan%20-%20Who's%20afraid%20of%20not%20completely%20positive%20maps%3F.pdf"

(Who’s afraid of not completely positive maps?) gave arguments that they do not exhaust all physical evolution possibilities.

Completely positive maps can be characterized without involving external systems by means of the Stinespring factorization theorem, which reduces to Choi's theorem for the case of finite dimensional Hilbert spaces:

$ \Phi(a) = \sum_{i=1}^{mn}V_i^\dagger a V_i$.

Completely positive maps are considered to represent the most general quantum evolutions, however, Shaji and Sudarshan in Who’s afraid of not completely positive maps? gave arguments that they do not exhaust all physical evolution possibilities.