Physics – Quantum Physics
Scientific paper
2002-01-30
Phys.Lett. A300 (2002) 107-114
Physics
Quantum Physics
11 pages, LaTeX, no figures. Further aspects of complete positivity worked out and references added; version accepted for publ
Scientific paper
10.1016/S0375-9601(02)00816-2
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 $D[\rho]$.
Bertlmann Reinhold A.
Grimus Walter
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