Volume fractions of the kinematic 'near-critical' sets of the quantum ensemble control landscape

An estimate is derived for the volume fraction of a subset C ^P_ε = {U : ∥ grad J(U) ∥ ≤ ε} ⊂ U(N) in the neighborhood of the critical set C^P ≃ U(n)PU(m) of the kinematic quantum ensemble control landscape J(U) = Tr(U _ρU†O), where U represents the unitary time evolution operator, ρ is the initial density matrix of the ensemble, and O is an observable operator. This estimate is based on the Hilbert.Schmidt geometry for the unitary group and a first-order approximation of ∥ grad J(U) ∥². An upper bound on these near-critical volumes is conjectured and supported by numerical simulation, leading to an asymptotic analysis as the dimension N of the quantum system rises in which the volume fractions of these 'enear-critical' sets decrease to zero as N increases. This result helps explain the apparent lack of influence exerted by the many saddles of J over the gradient flow.