Consider two quantum systems, Q and E where Q is some system of interest and E is some system that is external to Q and that is in some fixed pure state . Now let us suppose that the two systems interact and evolve via some unitary operator on the combined Hilbert space of each, . In this situation Q is known as an open system and E is the environment.
The dilation construction of quantum states (see Stinespring’s dilation theorem above), i.e. in the quantum operation formalism, the evolution of a system is often written in a more condensed manner as
Here we refer to as a superoperator.
Suppose is a linear map on Q-operators. Then the following three conditions are equivalent:
This is proven in Appendix D of
where there is also an explanation of “physically reasonable.”
Ian Durham: Is there a convenient category theoretic way to prove the above lemma?