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Definitions: a map $\Phi$ is called positive if $\Phi(\rho)$ is positive semidefinite for any positive semidefinite $\rho$, and completely positive (CP) if $\Phi \otimes \mathrm{Id}$ is a positive map with $\mathrm{Id}$ standing for an identity channel of arbitrary dimension.

It is well-known that a quantum channel needs to be a CP map in order to represent the physical evolution of a quantum system. However, I sometimes see mentions in the literature of a positive but not completely positive evolution of a system (and sometimes even beyond positive maps), in particular in the context of reduced dynamics of open quantum systems, e.g. [1], [2].

I am not too familiar with open quantum systems literature, but I would like to get some intuition for situations in which the physical evolution of a system might be expressed by a non-CP map. What classes of maps are used in such settings, and how is it all justified without "breaking physics"? Beyond the example of open quantum systems, does it make sense in any other context to allow a state to evolve under a non-CP map?

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