An open system in the sense you describe is not a system coupled to the environment, but the idealized version of this system where the environment has been eliminated using a Markov approximation, so that there is a closed dynamics on the system itself, without reference to the environment (and hence to measurement), and without memory (which is enforced by the Markov approximation). [To see how the environment is eliminated, see, e.g., cond-mat/0011204.]

Closed dynamics means that you can express the time derivative of the state of the system in terms of the current state and its past, and lack of memory means that you in fact have a differential equation for the states rather than an integrodifferental equation. (The above characterizxation would not work in the Heisenberg picture.)

If there are time-dependent external fields, the characterization you inquire about doesn't hold. But if there are no time-dependent external fields then the differential equation is autononous, and hence describes a semigroup.

For finitely many degrees of freedom, it is in fact a group, but running the dynamics backwards is highly unstable. For infinitely many degrees of freedom, running the dynamics backwards is ill-posed.

Thus the semigroup property has nothing to do with quantum mechanics (which just contributes complete positivity). Think of the heat equation as an example of a semigroup describing a dissipative (i.e., open) classical system.

But the form of the resulting dynamics is specifically quantum, typically a Lindblad equation; see, e.g., quant-ph/0302047.

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