Characterization of the critical submanifolds in quantum ensemble control landscapes

Abstract

A quantum control landscape is defined as the physical objective as a\nfunction of the control variables to be optimized. Analyzing the topology of\nthese landscapes is important for understanding the origins of the increasing\nnumber of laboratory successes in the optimal control of quantum processes.\nThis paper proposes a simple scheme to compute the characteristics of the\ncritical topology of the quantum ensemble control landscapes for observables,\nshowing that the set of disjoint critical submanifolds one-to-one corresponds\nto a finite number of contingency tables that solely depend on the degeneracy\nstructure of the eigenvalues of the initial system density matrix and the\nobservable to be controlled. The landscape characteristics can be calculated as\nfunctions of the table entries, including the dimensions and the numbers of\npositive and negative eigenvalues of the Hessian quadratic form of each of the\nconnected components of the critical submanifolds. Typical examples are given\nto illustrate the effectiveness of this method.\n