Mixed quantum states with variable Planck constant

•Several scientists have proposed that not only the fine structure constant but also Planck's constant could have varied since the Big Bang.•We discuss the effects of a variation of h on pure and mixed quantum systems.•We show that the purity of these states is very sensitive to changes in h.•The Gaussian case is studied in detail.•We discuss the difficulties inherent in the general case.

Recent cosmological measurements tend to confirm that the fine structure constant α is not immutable and has undergone a tiny variation since the Big Bang. Choosing adequate units, this could also reflect a variation of Planck's constant h. The aim of this Letter is to explore some consequences of such a possible change of h for the pure and mixed states of quantum mechanics. Surprisingly enough it is found that not only is the purity of a state extremely sensitive to such changes, but that quantum states can evolve into classical states, and vice versa. A complete classification of such transitions is however not possible for the moment being because of yet unsolved mathematical difficulties related to the study of positivity properties of trace class operators.