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Uniformly Stable Minimal Weyl--Heisenberg Measurements Approaching the SIC Benchmark — arXiv:2608.11850

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.11850
License (as archived): CC0-1.0
Archived: 2026-10-19T05:48:19.272Z
SHA-256 of the archived text: e52c49acceef0faedfd50adf5701ac0dda6899197531d3344a8347611a76635d
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Uniformly Stable Minimal Weyl--Heisenberg Measurements Approaching the SIC Benchmark — arXiv:2608.11850 — arXiv (en), https://arxiv.org/abs/2608.11850 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_fea656f3391c (Metraipolis archive dc_fea656f3391c, sha256 e52c49acceef0faedfd50adf5701ac0dda6899197531d3344a8347611a76635d)

BibTeX

@misc{dcfea656f3391c,
  title = {Uniformly Stable Minimal Weyl--Heisenberg Measurements Approaching the SIC Benchmark — arXiv:2608.11850},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.11850},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_fea656f3391c; sha256 e52c49acceef0faedfd50adf5701ac0dda6899197531d3344a8347611a76635d},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.11850. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 \(d|\chi_\phi(u)|^2\), turning stability into an explicit worst-direction design problem; write \(\lambda\) for its smallest nonidentity eigenvalue. Haar fiducials are IC almost surely while \(\mathbb E[\lambda^{-1}]=\infty\), 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 \(\Theta(d^{-3})\) for odd \(d\) and \(\Theta(d^{-5})\) for even \(d\); a finite-field family for \(q=2^m\) obeys the uniform bound \(\lambda\ge4/9\). 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 \(\lambda(\phi_q)/\Lambda_q^\star\to1\) for the global finite-field WH max--min optimum \(\Lambda_q^\star\), 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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-13 in math.IT).
Categorie arXiv: Quantum Physics (quant-ph) ; Information Theory (cs.IT).
Pagina del lavoro: https://arxiv.org/abs/2608.11850 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-19T05:41:22.363Z dalla pagina https://arxiv.org/catchup/math.IT/2026-08-13?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).