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

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.

• The paper reframes minimal Weyl–Heisenberg measurement design as a spectral max–min problem, showing that SIC measurements maximize the smallest nonidentity Gram eigenvalue and provide the stability benchmark.
• The authors construct explicit stable families, including a characteristic-two family with a uniform eigenvalue floor of 4/9 and balanced Alltop measurements whose normalized stability approaches the global finite-field optimum for characteristic p≥5.
• The results show that informational completeness alone does not ensure reliable tomography, while the spectral floor directly controls inverse amplification, worst-direction Fisher efficiency, shadow bounds, and canonical estimation error.

Overview and motivation

Informational completeness (IC) of a quantum measurement guarantees that a state can be recovered from outcome statistics, but it says nothing about the conditioning of the associated inverse problem. Zhu and Wang address this gap for minimal rank-one Weyl–Heisenberg (WH) measurements, where the projector-Gram spectrum is explicitly computable from the fiducial state's ambiguity function. Their central observation is that WH covariance diagonalizes the Gram matrix in the phase-space Fourier basis, with eigenvalues $d|\chi_\phi(u)|^2$, so stability reduces to controlling the smallest nonidentity ambiguity intensity $\lambda(\phi)$ — an explicit worst-direction design problem rather than an abstract frame-theoretic one.

The paper's organizing principle is the hierarchy

$$\text{SIC} \Longrightarrow \text{uniform spectral stability} \Longrightarrow \text{IC},$$

with all converses failing, and with balanced informationally complete (BIC) measurements sitting outside this hierarchy as a structural property. The SIC endpoint emerges not as an imposed symmetry target but as the max–min solution: since the Moyal identity fixes the total nonidentity spectral weight at $d^2-d$, the minimum eigenvalue cannot exceed the average $d/(d+1)$, with equality if and only if the orbit is a SIC-POVM.

The spectral interface and its operational content

The technical foundation is a restatement of the known eigensystem [Goldberger2022]: the two-dimensional discrete Fourier transform diagonalizes the projector-Gram matrix, and its eigenvalues are exactly the ambiguity intensities $d|\chi_\phi(u)|^2$ up to symplectic relabeling. This yields the dictionary that zeros mark loss of IC, small coefficients mark weakly resolved operator directions, and a flat nonidentity spectrum is the SIC endpoint.

Three operational consequences follow from this spectrum:

• Canonical inversion: the induced Hilbert–Schmidt norm of the inverse frame channel satisfies $\|\mathcal T_\phi^{-1}\|_{2\to2} = d/\lambda(\phi)$, so the weakest measurement direction is the most strongly amplified.
• Exact finite-sample tomography error: at the maximally mixed input, the Hilbert–Schmidt MSE of canonical linear inversion over $N$ snapshots equals $(1/N)\sum_{v\ne0}\lambda_v^{-1}$ — determined by the full spectrum, not just its edge. For a SIC this gives $(d^2-1)(d+1)/(Nd)$.
• Fisher information and shadows: the worst-direction classical-to-quantum Fisher-information ratio at $\lambda(\phi)$0 equals $\lambda(\phi)$1, and canonical-shadow second moments obey uniform bounds scaling as $\lambda(\phi)$2.

A notable quantitative contrast appears in the numerical benchmark: across displayed primes, the balanced-Alltop MSE ratio decreases from 1.474 to 1.194 relative to the SIC benchmark, while the Haar median rises from 3.117 to 6.108 and the geometric representative grows to $\lambda(\phi)$3. IC alone therefore does not control even fixed-state linear-inversion error.

Completeness without stability

Two separation results establish that generic or structural completeness does not imply stability.

Haar fiducials are IC almost surely, and each fixed nonidentity Gram eigenvalue has mean exactly $\lambda(\phi)$4 — the SIC value. Nevertheless, $\lambda(\phi)$5: rare near-singular realizations dominate the inverse. The proof uses beta-distribution behavior in even dimension and a coarea argument on the probability simplex in odd dimension. Consequently, the paper's Haar benchmarks are reported via quantiles of $\lambda(\phi)$6 rather than means of $\lambda(\phi)$7.

Geometric BIC constructions fare worse. The truncated geometric fiducial used by Farkas et al. to prove BIC existence in every dimension has a SIC-normalized floor bounded above by $\lambda(\phi)$8 — exponential decay. The paper also supplies a complete phase classification: in even dimension with $\lambda(\phi)$9, the orbit has exactly $$\text{SIC} \Longrightarrow \text{uniform spectral stability} \Longrightarrow \text{IC},$$0 zero Gram eigenvalues; in particular, $$\text{SIC} \Longrightarrow \text{uniform spectral stability} \Longrightarrow \text{IC},$$1 fails IC whenever $$\text{SIC} \Longrightarrow \text{uniform spectral stability} \Longrightarrow \text{IC},$$2. These results do not affect the ideal device-independent randomness certification of BIC structure, which concerns exact properties at maximal Bell violation.

Against these negative results, an explicit parity-dependent cyclic family achieves polynomial floors in every integer dimension: $$\text{SIC} \Longrightarrow \text{uniform spectral stability} \Longrightarrow \text{IC},$$3 for odd $$\text{SIC} \Longrightarrow \text{uniform spectral stability} \Longrightarrow \text{IC},$$4 (with exact formula $$\text{SIC} \Longrightarrow \text{uniform spectral stability} \Longrightarrow \text{IC},$$5) and $$\text{SIC} \Longrightarrow \text{uniform spectral stability} \Longrightarrow \text{IC},$$6 for even $$\text{SIC} \Longrightarrow \text{uniform spectral stability} \Longrightarrow \text{IC},$$7, giving $$\text{SIC} \Longrightarrow \text{uniform spectral stability} \Longrightarrow \text{IC},$$8 universally.

Uniformly stable finite-field families

For $$\text{SIC} \Longrightarrow \text{uniform spectral stability} \Longrightarrow \text{IC},$$9, a two-component fiducial $d^2-d$0 with phase schedule depending on $d^2-d$1 yields a complete closed-form spectrum with three branches. The floor obeys $d^2-d$2 uniformly, covering every multi-qubit Hilbert-space dimension, with $d^2-d$3. At $d^2-d$4, all 63 nonidentity eigenvalues coalesce at $d^2-d$5, attaining the SIC endpoint exactly, as does $d^2-d$6. However, the branch ratio grows as $d^2-d$7, so this family is uniformly stable but not asymptotically spectrally flat.

Balanced Alltop: asymptotic optimality without SIC existence

The main result concerns prime powers of characteristic $d^2-d$8. The cubic Alltop state has an ambiguity profile flat at magnitude $d^2-d$9 off one axis, but vanishes on that axis ($d/(d+1)$0 zero Gram eigenvalues). A single-coordinate spike $d/(d+1)$1 repairs the axis: the repaired-axis amplitude scales as $d/(d+1)$2 while bulk distortion is linear in $d/(d+1)$3. Balancing these competing scales selects $d/(d+1)$4.

At this balance point, the main theorem establishes that the exact floor is attained on the repaired axis,

$d/(d+1)$5

the entire nonidentity spectrum lies in $d/(d+1)$6 with $d/(d+1)$7, and $d/(d+1)$8 uniformly. Asymptotically, $d/(d+1)$9, so $d|\chi_\phi(u)|^2$0.

The strongest claim is the max–min squeeze: writing $d|\chi_\phi(u)|^2$1 for the global finite-field WH optimum, Proposition 2 and the theorem give

$d|\chi_\phi(u)|^2$2

so $d|\chi_\phi(u)|^2$3 without assuming SIC existence, with the quantitative gap $d|\chi_\phi(u)|^2$4. This is an unconditional asymptotic optimality statement — though it does not identify any finite-$d|\chi_\phi(u)|^2$5 global optimizer, and $d|\chi_\phi(u)|^2$6 is provably not a SIC at finite $d|\chi_\phi(u)|^2$7.

Operationally, the traceless condition number tends to one as $d|\chi_\phi(u)|^2$8, pairwise overlaps satisfy $d|\chi_\phi(u)|^2$9, and both Fisher-efficiency endpoints and shadow-bound overheads converge to their SIC values after normalization. The physical scaled-frame minimum remains $\|\mathcal T_\phi^{-1}\|_{2\to2} = d/\lambda(\phi)$0, matching the unavoidable dimensional scaling of a SIC; the gain is a dimension-independent relative factor plus vanishing anisotropy.

Limitations and open questions

The paper is explicit about boundaries. The cubic mechanism excludes characteristics two and three; characteristic two is handled separately, but a uniformly stable finite-field construction in characteristic three remains open. Finite-field results use the group $\|\mathcal T_\phi^{-1}\|_{2\to2} = d/\lambda(\phi)$1, distinct from cyclic $\|\mathcal T_\phi^{-1}\|_{2\to2} = d/\lambda(\phi)$2 when $\|\mathcal T_\phi^{-1}\|_{2\to2} = d/\lambda(\phi)$3, so they do not extend the cyclic construction. The characteristic-two fiducial provides no efficient measurement circuit. The spike size $\|\mathcal T_\phi^{-1}\|_{2\to2} = d/\lambda(\phi)$4 optimizes the certified analytic lower bound, not a proved finite-$\|\mathcal T_\phi^{-1}\|_{2\to2} = d/\lambda(\phi)$5 global objective within the one-spike family, let alone the true max–min optimum. The exact-MSE corollary is scoped to the maximally mixed input, independent samples, canonical inversion, and Hilbert–Schmidt norm. Finally, whether the single-frame spectral floor predicts robustness of BIC-based certification away from ideal maximal violation, and whether augmented frames improve canonical-estimator guarantees, are left unanswered.

Conclusion

This work reframes minimal WH measurement design as an explicit spectral max–min problem whose endpoint is the SIC, and constructs a hierarchy of explicit solutions: polynomial floors in every integer dimension, a uniform floor of $\|\mathcal T_\phi^{-1}\|_{2\to2} = d/\lambda(\phi)$6 across all multi-qubit dimensions, and, for characteristic $\|\mathcal T_\phi^{-1}\|_{2\to2} = d/\lambda(\phi)$7, a balanced-Alltop family whose full nonidentity spectrum collapses isotropically onto the SIC value while approaching the global finite-field WH optimum unconditionally. The accompanying operational analysis ties the Gram edge to inverse amplification, worst-direction Fisher efficiency, and shadow bounds, and the full spectrum to exact canonical tomography error — establishing that informational completeness is only the starting point, and that spectral design determines whether minimal measurements remain statistically useful as dimension grows.