Persistence of quantum information

We study the flip-processes in a two-level system, which is triggered by the coupling to a classical bath. When the bath is represented by a stochastic field, the time evolution of the density matrix leads to a stochastic equation with a multiplicative noise. Accordingly the Fokker-Planck-equation (FPE) depends on the matrix elements of the underlying density operator. The solution of the FPE can be parametrized in terms of an inherent conserved quantity α, which is interpreted as a measure for the persistence of quantum information. We show that the FPE exhibits a single unique steady state solution different from Boltzmann's law. The exactly computable discrete spectrum of the relaxation times is characterized by two quantum numbers and the ratio of Planck's constant and the coupling strength to the bath. The total entropy is analyzed as function of the quantum number α. In case of α=1 the system is in a pure state whereas for α≠1 a mixed state is realized. In case of two, two-level systems, immersed in the common bath, the two noninteracting two-level systems become mutually entangled. The annealed entropy is in that case non-extensive.