Control of Finite-Dimensional Quantum Systems under Lindblad Dissipation.
Darragh Patrick Rooney · Deep Blue (University of Michigan) · 2012
This thesis considers the problem of controlling finite-dimensional quantum density matrices under the influence of Lindblad dissipation. The primary insight is that, when one assumes arbitrary Hamiltonian control, we can project the Lindblad equation onto a set of differential equations over the set of orbits. These new control equations are valid only on the interior of the orbit set, but we have shown that controllability results on enlarged sets can safely be applied to this interior. We provide a thorough controllability analysis for arbitrary two-dimensional systems and show that every system has a trap, inside of which one has full controllability, but from which a system cannot escape. We also provide a result classifying purifiable two-dimensional systems. For higher dimensional systems, we can also project the Lindblad equation to differential equations on the interior of the set of orbits. We have studied a reduced control problem where one considers only a set of $n!$ controls that are generated by the symmetric group $S_n$. We can construct sets of small-time local controllability for arbitrary dimensions, and have provided some results on global controllability for three dimensions. Finally, we have provided two combinatorial formulas relating certain points on the STLC set for three dimensions to rooted trees. One of these formulas has been shown to generalize to higher dimensions.