Non-completely Positive Maps
Shaji and Sudarshan argued in a Physics Letters A paper that upon close examination complete positivity is not a requirement for a good representation of open quantum evolution. Their calculations show that if we start with some fixed initial correlations between the observed system and the environment then the map restricted to the system itself is not necessarily even positive. However, it is not positive only for those states that do not satisfy the assumption about the form of initial correlations. Thus, they show that to get a full understanding of quantum evolution, non completely-positive maps should be considered as well
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