Topological properties of reachable sets and the control of quantum bits

Claudio Altafini

Haidong Yuan

E. O. Ayoola

We establish an exponential formula for the reachable sets of quantum stochastic differential inclusions (QSDI) which are locally Lipschitzian with convex values. Our main results partially rely on an auxilliary result concerning the density, in the topology of the locally convex space of solutions, of the set of trajectories whose matrix elements are continuously differentiable. By applying the exponential formula, we obtain results concerning convergence of the discrete approximations of the reachable set of the QSDI. This extends similar results of Wolenski[20] Wolenski, P.R. 1990. The exponential formula for the reachable set of a Lipschitz differential inclusion. SIAM J. Control Optim., 28(5): 1148–1161. [Crossref], [Web of Science ®] , [Google Scholar] for classical differential inclusions to the present noncommutative quantum setting.

Louis Hirsch Kauffman, Samuel J. Lomonaco

Yangjia Li, Mingsheng Ying

Lev Vyacheslavovich Lokutsievskiy, Alexander Nikolaevich Pechen

Abstract In this work, we study controllability in the set of all density matrices for a two-level open quantum system driven by coherent and incoherent controls. In Pechen (2011 Phys. Rev. A 84 042106) an approximate controllability, i.e. controllability with some precision, was shown for generic N -level open quantum systems driven by coherent and incoherent controls. However, the explicit formulation of this property, including the behavior of this precision as a function of transition frequencies and decoherence rates of the system, was not known. The present work provides a rigorous analytical study of reachable sets for two-level open quantum systems. First, it is shown that for N = 2 the presence of incoherent control does not affect the reachable set (while incoherent control may affect the time necessary to reach particular state). Second, the reachable set in the Bloch ball is described and it is shown that already just for one coherent control any point in the Bloch ball can be achieved with precision δ ∼ γ / ω , where γ is the decoherence rate and ω is the transition frequency. Typical values are δ ≲ 10 −3 that implies high accuracy of achieving any density matrix. Moreover, we show that most points in the Bloch ball can be exactly reached, except of two lacunae of size ∼ δ . For two coherent controls, the system is shown to be completely controllable in the set of all density matrices. Third, the reachable set as a function of the final time is found and shown to exhibit a non-trivial structure.