= Quantum dynamical semigroup
A <quantum dynamical semigroup> is a continuous family $(\Phi_t)_{t\ge0}$ of linear maps on <density matrices> satisfying $\Phi_0=I$ and $\Phi_{t+s}=\Phi_t\circ\Phi_s$, with each map <completely positive> and trace preserving. The composition law expresses time-homogeneous evolution with no dependence on how the interval is divided.
In finite dimensions, continuity gives $\Phi_t=e^{t\mathcal L}$, whose generator has <Lindblad equation> form. For a qubit dephasing example,
$$
\Phi_t\!\begin{pmatrix}a&b\\b^*&1-a\end{pmatrix}=\begin{pmatrix}a&e^{-\gamma t}b\\e^{-\gamma t}b^*&1-a\end{pmatrix},\qquad\gamma\ge0.
$$
Off-diagonal decay factors multiply under composition, while populations stay fixed. The map is the mixture $(1+e^{-\gamma t})\rho/2+(1-e^{-\gamma t})Z\rho Z/2$, which verifies <complete positivity>. A semigroup need not have physical inverse maps at positive times: undoing this decay would generally amplify coherence beyond valid states.<On the generators of quantum dynamical semigroups by G. Lindblad>[Generator characterization]{ref}
Back to article page
