**Quantum brachistochrone problem for spin-1/2 mixed states with decoherence**

**Abstract:**

We study passage time needed for the evolution of the initial mixed spin-1/2 state into final one in magnetic field. The time of evolution
depends only on the angle between the Bloch vectors of initial and final states and the angle of precession. Minimal time
of evolution that corresponds to the quantum brachistochrone is achieved when magnetic field is perpendicular to the Bloch vector. We also take into
account decoherence, which is caused by the
fluctuations of the absolute value of magnetic field with fixed direction. Considering the quantum brachistochrone problem we characterize initial and
final states only by the direction of the Bloch vector. It is interesting that when magnetic field is
perpendicular to the initial Bloch vector the decoherence does not effect on the passage time. It is shown that in linear approximation over the parameter
of decoherence the passage time is lager than without decoherence. In this approximation the minimal time is equal to the minimal time of evolution without
decoherence and is achieved when the magnetic field is
perpendicular to the initial Bloch vector.