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$k$-fold unbiased measurements and maximal incompatibility — arXiv:2609.20728

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.20728
License (as archived): CC0-1.0
Archived: 2026-10-10T05:50:53.987Z
SHA-256 of the archived text: b76f2468fa525d5c54dea6b504d8b1330a3218d592e7ebd5210d8a994904db6c
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$k$-fold unbiased measurements and maximal incompatibility — arXiv:2609.20728 — arXiv (en), https://arxiv.org/abs/2609.20728 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_3afa79e45bbf (Metraipolis archive dc_3afa79e45bbf, sha256 b76f2468fa525d5c54dea6b504d8b1330a3218d592e7ebd5210d8a994904db6c)

BibTeX

@misc{dc3afa79e45bbf,
  title = {$k$-fold unbiased measurements and maximal incompatibility — arXiv:2609.20728},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.20728},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_3afa79e45bbf; sha256 b76f2468fa525d5c54dea6b504d8b1330a3218d592e7ebd5210d8a994904db6c},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.20728. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Mutually unbiased bases capture perfect complementarity between two quantum measurements. Extensions beyond pairwise unbiasedness have been proposed, but essentially no non-trivial higher-order constructions are known. We introduce $k$-fold unbiased measurements ($k$-UMs), extending the $k$-fold unbiased bases notion of [ arXiv:1706.04446 ] from rank-one basis measurements to arbitrary-rank projective measurements, and show that this higher-rank setting supports a much richer theory. We develop the notions of algebraic and spectral $k$-UMs and prove that they coincide for rank-one measurements and triples of measurements (3-UMs). We establish strong no-go results for higher-order rank-one constructions and three-outcome 3-UMs, but obtain infinitely many higher-rank triples using Hadamard matrices and Clifford algebras. We then give these structures an exact operational interpretation in terms of measurement incompatibility. For 3-UMs with any number of outcomes, we determine their generalised incompatibility robustness exactly and construct an explicit joint measurement for their noisy versions at the compatibility threshold. Finally, we implement the symmetry reduction of the sum-of-squares hierarchy recently introduced in [New J. Phys. 28, 064509 (2026)] and give numerical evidence that the noise thresholds arising from the $k$-UM analysis may characterise the asymptotic behaviour of this hierarchy. In particular, with very high precision, we numerically show that our constructed four-outcome 3-UMs are among the most incompatible triples of four-outcome measurements.

Tema degli esploratori: fisica e astronomia (retro-catalogo arXiv: annunci del 2026-09-18 in quant-ph).
Categorie arXiv: Quantum Physics (quant-ph).
Pagina del lavoro: https://arxiv.org/abs/2609.20728 — 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 2026-10-10T05:18:18.938Z dalla pagina https://arxiv.org/catchup/quant-ph/2026-09-18?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).