Inherently trap-free convex landscapes for fully quantum optimal control

A general quantum system may be steered by a control of either classical or quantum nature and the latter scenario is particularly important in many quantum engineering problems including coherent feedback and reservoir engineering. In this paper, we consider a quantum system steered by a quantum controller and explore the underlying Q–Q (quantum–quantum) control landscape features for the expectation value of an arbitrary observable of the system, with the control being the engineered initial state of the quantum controller. It is shown that the Q–Q control landscape is inherently convex, and hence devoid of local suboptima. Distinct from the landscapes for quantum systems controlled by time-dependent classical fields, the controllability is not a prerequisite for the Q–Q landscape to be trap-free, and there are no saddle points that generally exist with a classical controller. However, the forms of Hamiltonian, the flexibility in choosing initial state of the controller, as well as the control duration, can influence the reachable optimal value on the landscape. Moreover, we show that the optimal solution of the Q–Q control landscape can be readily extracted from a de facto landscape observable playing the role of an effective “observer”. For illustration of the basic Q–Q landscape principles, we consider the Jaynes–Cummings model depicting a two-level atom in the presence of a cavity quantized radiation field.