On controllability of some quantum systems

Consider dynamics of a quantum system driven by some controls (external actions) which belong to some admissible set. If the system is closed, then it is called completely controllable if any unitary operator can be realized as an evolution with some admissible control. If the system is open, then it is called controllable in the set of density matrices if any initial density matrix can evolve into any target density matrix under some admissible controls. If the system is uncontrollable, its reachable sets are the sets of density matrices which can be obtained starting from a given initial state and using all admissible controls of arbitrary duration. Controllability and reachable sets are among the main properties which should be established when studying a given controlled quantum system. In this talk, we will discuss some results about controllability of closed four-level systems with degenerate Hamiltonians, which are controllable or not depending on values of some parameters in the Hamiltonian [1,2], and about controllability in the set of density matrices and reachable sets for a qubit interacting with the environment and driven by coherent and incoherent controls, where there exist unreachable sets of points in the Bloch ball [3]. Finally we discuss the use of speed gradient approach for finding optimal coherent and incoherent controls for energy transfer in a quantum oscillator interacting with the environment. Partial support Priority 2030, K2-2022-025, RSF 22-11-00330.