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A quantum control landscape is defined as the physical objective as afunction of the control variables to be optimized. Analyzing the topology ofthese landscapes is important for understanding the origins of the increasingnumber of laboratory successes in the optimal control of quantum processes.This paper proposes a simple scheme to compute the characteristics of thecritical topology of the quantum ensemble control landscapes for observables,showing that the set of disjoint critical submanifolds one-to-one correspondsto a finite number of contingency tables that solely depend on the degeneracystructure of the eigenvalues of the initial system density matrix and theobservable to be controlled. The landscape characteristics can be calculated asfunctions of the table entries, including the dimensions and the numbers ofpositive and negative eigenvalues of the Hessian quadratic form of each of theconnected components of the critical submanifolds. Typical examples are givento illustrate the effectiveness of this method.

Characterization of the Critical Submanifolds in Quantum Ensemble Control Landscapes