  We address the following inverse problem in quantum statistical physics: does
the quantum free energy (von Neumann entropy + kinetic energy) admit a unique
minimizer among the density operators having a given local density $n(x)$? We
give a positive answer to that question, in dimension one. This enables to
define rigourously the notion of local quantum equilibrium, or quantum
Maxwellian, which is at the basis of recently derived quantum hydrodynamic
models and quantum drift-diffusion models. We also characterize this unique
minimizer, which takes the form of a global thermodynamic equilibrium
(canonical ensemble) with a quantum chemical potential.
