Dynamics of a set of quantum states generated by a nonlinear Liouville–von Neumann equation

A model describing the dynamics of a set of quantum states generated by a nonlinear Schrödinger equation is studied. The relationship between the blow-up of a solution with self-focusing and the transition from pure to mixed states of a quantum system was investigated in [1]. In this context, a natural question is concerned with the dynamics generated by the nonlinear Schrödinger equation in the set of mixed quantum states. The dynamics of mixed quantum states is described by the Liouville–von Neumann equation corresponding to the nonlinear Schrödinger equation. For the former equation, conditions for the global existence of a unique solution of the Cauchy problem and blow-up conditions are obtained.