Quantum dynamical semigroup

In quantum mechanics, and especially the theory of open quantum systems, a quantum dynamical semigroup is family of quantum dynamical maps $\phi _{t}$ on the space of density matrices indexed by a single time parameter $t\geq 0$ that obey the semigroup property

\phi_s(\phi_t(\rho)) = \phi_{t+s}(\rho) , \qquad t,s \ge 0.

A quantum dynamical semigroup is generated by a Lindblad superoperator. The generator can be obtained by

\mathcal{L}(\rho) = \mathrm{lim}_{\Delta t \to 0} \frac{\phi_{\Delta t}(\rho)-\phi_0(\rho)}{\Delta t}

which, by the linearity of $\phi _{t}$ , is a linear superoperator. The semigroup can be recovered as

\phi_{t+s}(\rho) = e^{\mathcal{L}s} \phi_t(\rho).

References

Lindblad, G. (1976). "On the generators of quantum dynamical semigroups". Commun. Math. Phys. 48 (2): 119. Bibcode:1976CMaPh..48..119L. doi:10.1007/BF01608499.
Alicki, Robert. "Invitation to quantum dynamical semigroups". arXiv:quant-ph/0205188.
Alicki, Robert; Lendi, Karl (1987). Quantum Dynamical Semigroups and Applications. Berlin: Springer Verlag. ISBN 978-0-3871-8276-6.

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