Blow-Up of States in the Dynamics Given by the Schrödinger Equation with a Power-Law Nonlinearity in the Potential

We study specific features of the dynamics of quantum states generated by the focusing nonlinear Schrödinger equation. Relationships between phenomena of the self-focusing, unbounded growth in the solution gradient, and transition from a pure to a mixed quantum state are established. It is shown that the continuation of the dynamics of a pure quantum state beyond the gradient blow-up time can be realized using a partial trace of the nonlinear dynamics of a pure state in an extended Hilbert space.