Control landscapes for open system quantum operations

The reliable realization of control operations is a key component of quantum information applications. In practice, meeting this goal is very demanding for open quantum systems. This paper investigates the landscape defined as the fidelity J between the desired and achieved quantum operations with an open system. The goal is to maximize J as a functional of the control variables. We specify the complete set of critical points of the landscape function in the so-called kinematic picture. An associated Hessian analysis of the landscape reveals that, upon the satisfaction of a particular controllability criterion, the critical topology is dependent on the particular environment, but no false traps (i.e. suboptimal solutions) exist. Thus, a gradient-type search algorithm should not be hindered in searching for the ultimate optimal solution with such controllable systems. Moreover, the maximal fidelity is proven to coincide with Uhlmann's fidelity between the environmental initial states associated with the achieved and desired quantum operations, which provides a generalization of Uhlmann's theorem in terms of Kraus maps.