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

Abstract

We consider the time evolution of the density matrix ρ in a 2-dimensional complex Hilbert space. We allow for dissipation by adding to the von Neumann equation a term D[ ρ], which is of Lindblad type in order to assure complete positivity of the time evolution. We present five equivalent forms of D[ ρ]. In particular, we connect the familiar dissipation matrix L with a geometric version of D[ ρ], where L consists of a positive sum of projectors onto planes in R 3 . We also study the minimal number of Lindblad terms needed to describe the most general case of D[ ρ]. 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[ ρ]. Finally, we perform a general discussion of the asymptotic behaviour t→∞ of the density matrix and we relate the two types of asymptotic behaviour with our geometric version of D[ ρ].