Comment on "Are there traps in quantum control landscapes?".

Many quantum control problems are formulated as a search for an optimal field that maximizes a physical objective. This search is performed over a landscape defined as the objective as a function of the control field. A recent Letter [1] asserts that the existence of special landscape critical points (CPs) with trapping characteristics is " contrary to recent claims in the literature " and " can have profound implications for both theoretical and experimental quantum control studies. " We show here that these assertions are inaccurate and misleading. First, the authors in [1] declare that their finding of local traps at singular CPs corresponding to constant control fields contradicts the existing theory of quantum control landscapes. However, in making this statement, they completely focus on an early paper published in 2004 [2] and improperly ignore the large body of subsequent research, in which a careful distinction has been made between regular and singular CPs (see Refs. [3, 4] for reviews). Specifically, the current status of the theory is that the equivalence of kinematic and dynamic control landscapes and the resulting absence of local traps applies only to regular CPs (at which the tangent map from the space of controls to the space of evolution operators is locally surjective) [4]. No such assessment is made in the literature for singular CPs (including constant-field solutions reported in [1]). Already in 2007, Chakrabarti and Rabitz [3] discussed the existence of singular CPs at constant control fields and clearly stated that the results regarding the trap-free character of control landscapes apply only to regular CPs. This distinction between regular and singular CPs and the applicability of the trap-free result only to regular ones was further stated in a number of subsequent works [4, 5]. With regard to singular CPs, it was only assumed that their measure in the search space is much smaller than that of regular ones, and this assumption is fully supported by the literature (see below). Correspondingly, no specific assessment is made in the current landscape theory with regard to the optimality of singular CPs, hence the report in [1] of locally optimal singular controls in no way contradicts this theory. Second, the statement in [1] with regard to implications of their work for quantum control studies is not supported by any evidence. In fact, the singular CPs reported in [1] are special cases of constant (e.g., zero) fields with strictly …