Dissipation in a 2-dimensional Hilbert space: Various forms of complete positivity

Abstract

We consider the time evolution of the density matrix $\rho$ in a 2-dimensional complex Hilbert space. We allow for dissipation by adding to the von Neumann equation a term $D[\rho ]$, which is of Lindblad type in order to assure complete positivity of the time evolution. We present five equivalent forms of $D[\rho ]$. In particular, we connect the familiar dissipation matrix $L$ with a geometric version of $D[\rho ]$, where $L$ consists of a positive sum of projectors onto planes in ${\mathbf{R}}^3$. We also study the minimal number of Lindblad terms needed to describe the most general case of $D[\rho ]$. All proofs are worked out comprehensively, as they present at the same time a practical procedure how to determine explicitly the different forms of $D[\rho ]$. Finally, we perform a general discussion of the asymptotic behaviour $t\to \infty$ of the density matrix and we relate the two types of asymptotic behaviour with our geometric version of $D[\rho ]$.

Comments: 11 pages, LaTeX, no figures. Further aspects of complete positivity worked out and references added; version accepted for publication in Phys. Lett. A