Singularity of dynamical maps
S. C. Hou, X. X. Yi, Sixia Yu, C. H. Oh · Physical Review A · 2012
For a dynamical map $\ensuremath{\Lambda}(t,0)$, which sends a state $\ensuremath{\rho}(0)$ of a quantum open system to a state $\ensuremath{\rho}(t)=\ensuremath{\Lambda}(t,0)\ensuremath{\rho}(0)$, the decomposition law $\ensuremath{\Lambda}(t,0)=\ensuremath{\Lambda}(t,{t}_{c})\ensuremath{\Lambda}({t}_{c},0)$ 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, and the measure of singularity for these singular points is calculated and discussed. An extension to a high-dimensional system is presented.