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.