  An estimate is derived for the volume fraction of a subset $C_{\epsilon}^{P}
= \{U : ||grad J(U)|\leq {\epsilon}\}\subset\mathrm{U}(N)$ in the neighborhood
of the critical set $C^{P}\simeq\mathrm{U}(\mathbf{n})P\mathrm{U}(\mathbf{m})$
of the kinematic quantum ensemble control landscape J(U) = Tr(U\rho U' O),
where $U$ represents the unitary time evolution operator, {\rho} 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 "near-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.
