Singularity of dynamical maps

For a dynamical map $Λ(t,0)$, which sends a state $ρ(0)$ of quantum open system to a state $ρ(t)=Λ(t,0)ρ(0)$, the decomposition law $Λ(t,0)=Λ(t,t_c)Λ(t_c,0)$ may break down at a specific time $t_c$. In this paper, we present a method to find the singular points $t_c$ and propose a measure for the singularity of the dynamical map. Two examples are portrayed to illustrate the method, the measure of singularity for these singular points is calculated and discussed. An extension to high-dimensional system is presented.

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