Reachability in Controlled Markovian Quantum Systems: An Operator-Theoretic Approach

In quantum systems theory one of the fundamental problems boils down to: Given an initial state, which final states can be reached by the dynamic system in question? Formulated in the framework of bilinear control systems, the evolution shall be governed by an inevitable Hamiltonian drift term, finitely many control Hamiltonians allowing for (at least) piecewise constant control amplitudes, plus a (possibly bang-bang switchable) noise term in Kossakowski-Lindblad form. Now assuming switchable coupling of finite-dimensional systems to a thermal bath of arbitrary temperature, the core problem of reachability boils down to studying points in the standard simplex amenable to two types of controls that can be used interleaved: Permutations within the simplex, and contractions by a dissipative one-parameter semigroup. We illustrate how the solutions of the core problem pertain to the reachable set of the original controlled Markovian quantum system. This allows us to show that for global as well as local switchable coupling to a temperature-zero bath one can generate every quantum state from every initial state up to arbitrary precision. Moreover we present an inclusion for non-zero temperatures as a consequence of our results on d-majorization. Then we consider infinite-dimensional open quantum-dynamical systems following a unital Kossakowski-Lindblad master equation extended by controls. Here the drift Hamiltonian can be arbitrary, the finitely many control Hamiltonians are bounded, and the switchable noise term is generated by a single compact normal operator. Via new majorization results of ours, we show that such bilinear quantum control systems allow to approximately reach any target state majorized by the initial one, as up to now only has been known in finite-dimensional analogues.