Quantum Systems Theory Viewed from Kossakowski-Lindblad Lie Semigroups - and Vice Versa

The solutions to the celebrated Kossakowski-Lindblad equation extended by coherent controls yield Markovian quantum maps. More precisely, the set of all its solutions forms a semigroup of completely positive trace-preserving maps taking the specific form of a Lie semigroup. Non-trivial symmetries of these semigroups are shown to preclude accessibility in Markovian dissipative systems. This is the open-system analogue to closed systems, where triviality of (quadratic) symmetries of the Hamiltonian part suffices to decide that the system is fully controllable. The findings are placed into a unifying Lie frame of quantum systems and control theory alongside with illustrating examples.

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