Uniformly Stable Minimal Weyl--Heisenberg Measurements Approaching the SIC Benchmark

Informational completeness (IC) guarantees that an inverse exists, not that it is statistically well conditioned. For minimal rank-one Weyl–Heisenberg (WH) measurements, covariance makes the nonidentity projector-Gram spectrum proportional to the fiducial’s ambiguity intensities, with eigenvalues $d\mathrm{\mid }{\chi }_{\varphi }(u){\mathrm{\mid }}^2$, turning stability into an explicit worst-direction design problem; write $\lambda$ for its smallest nonidentity eigenvalue. Haar fiducials are IC almost surely while $\mathbb{E}[{\lambda }^{-1}]=\mathrm{\infty }$, and an explicit geometric family used to establish balanced informationally complete measurements in every dimension has a normalized spectral floor bounded above by an exponentially decaying envelope. We then construct a hierarchy of minimal measurements. A cyclic family with exactly $d^2$ outcomes in every integer dimension has floors $\mathrm{\Theta }(d^{-3})$ for odd $d$ and $\mathrm{\Theta }(d^{-5})$ for even $d$; a finite-field family for $q=2^m$ obeys the uniform bound $\lambda \ge 4\mathrm{/}9$. Our main result treats every prime-power dimension of characteristic $p\ge 5$. A balanced one-coordinate perturbation repairs the zero ambiguity axis of a cubic Alltop state, gives an attained floor uniformly bounded below by a positive constant, and confines the entire nonidentity spectrum to $[L_q,U_q]$ with $U_q\mathrm{/}{L}_q\to 1$. Its SIC-normalized minimum tends to one, and $\lambda ({\varphi }_q)\mathrm{/}{\mathrm{\Lambda }}_q^{\star }\to 1$ for the global finite-field WH max–min optimum ${\mathrm{\Lambda }}_q^{\star }$, without assuming SIC existence. The complete spectrum determines the exact finite-sample Hilbert–Schmidt error of canonical linear inversion at $I\mathrm{/}d$, while its lower edge controls local Fisher efficiency and canonical-shadow bounds.