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

• URL: http://arxiv.org/abs/2608.11850v1
• Date: Wed, 12 Aug 2026 09:41:26 GMT
• Title: Uniformly Stable Minimal Weyl--Heisenberg Measurements Approaching the SIC Benchmark
• Abstract summary: Informational completeness (IC) guarantees that an inverse exists, not that it is statistically well conditioned.<n>For minimal rank-one Weyl--Heisenberg measurements, covariance makes the nonidentity projector-Gram spectrum proportional to the fiducial's ambiguity intensities.<n>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/L_qto1
• Score: 2.5735476569508995
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 \(d|χ_φ(u)|^2\), turning stability into an explicit worst-direction design problem; write \(λ\) for its smallest nonidentity eigenvalue. Haar fiducials are IC almost surely while \(\mathbb E[λ^{-1}]=\infty\), 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 \(Θ(d^{-3})\) for odd \(d\) and \(Θ(d^{-5})\) for even \(d\); a finite-field family for \(q=2^m\) obeys the uniform bound \(λ\ge4/9\). Our main result treats every prime-power dimension of characteristic \(p\ge5\). 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 \(U_q/L_q\to1\). Its SIC-normalized minimum tends to one, and \(λ(φ_q)/Λ_q^\star\to1\) for the global finite-field WH max--min optimum \(Λ_q^\star\), without assuming SIC existence. The complete spectrum determines the exact finite-sample Hilbert--Schmidt error of canonical linear inversion at \(I/d\), while its lower edge controls local Fisher efficiency and canonical-shadow bounds.

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