  We consider in this work the problem of minimizing the von Neumann entropy
under the constraints that the density of particles, the current, and the
kinetic energy of the system is fixed at each point of space. The unique
minimizer is a self-adjoint positive trace class operator, and our objective is
to characterize its form. We will show that this minimizer is solution to a
self-consistent nonlinear eigenvalue problem. One of the main difficulties in
the proof is to parametrize the feasible set in order to derive the
Euler-Lagrange equation, and we will proceed by constructing an appropriate
form of perturbations of the minimizer. The question of deriving quantum
statistical equilibria is at the heart of the quantum hydrody-namical models
introduced by Degond and Ringhofer in [5]. An original feature of the problem
is the local nature of constraints, i.e. they depend on position, while more
classical models consider the total number of particles, the total current and
the total energy in the system to be fixed.
