Why do effective quantum controls appear easy to find?

Experimental evidence shows that effective quantum controls in diverse applications appear surprisingly easy to find. The underlying reasons for this attractive behavior are explored in this work through an examination of the quantum control landscape of 〈 O ( T ) 〉 = Tr ( ρ ( T ) O ) directly in terms of the physically relevant control field ɛ ( t ) and the density matrix ρ ( T ) at the target time T, including an elaboration of the topology around the critical points, where δ 〈 O ( T ) 〉 / δ ɛ ( t ) = 0 ∀ t , of an arbitrary physical observable O . It is found that for controllable quantum systems the critical points of the landscape 〈 O ( T ) 〉 correspond to the global maximum and minimum and intermediate saddle points of 〈 O ( T ) 〉 . An upper bound is shown to exist on the norm of the slope δ 〈 O ( T ) 〉 / δ ɛ ( t ) anywhere over the landscape, implying that the control landscape has gentle slopes permitting stable searches for optimal controls. Moreover, the Hessian at the global maximum (minimum) only possesses a finite number of negative (positive) non-zero eigenvalues and the sum of the corresponding eigenvalues is bounded from below (above). The number of negative eigenvalues of the Hessians evaluated at the saddle points drops as the critical point value 〈 O ( T ) 〉 becomes smaller and finally converts to all positive non-zero eigenvalues at the global minimum. Collectively, these findings reveal that (a) there are no false traps at the sub-optimal extrema in the landscape, (b) the searches for optimal controls should generally be stable, and (c) an inherent degree of robustness to noise exists around the global optimal control solutions. As a result, it is anticipated that effective control over quantum dynamics may be expected even in highly complex systems provided that the control fields are sufficiently flexible to traverse the associated landscape.