Analytic construction of dimension-dependent steering witnesses from generalized equiangular measurements
Abstract
We construct a dimension-dependent family of generalized equiangular measurements that gives explicit steering witnesses in every prime dimension d≥5. The construction consists of d−1 mutually unbiased measurements and an additional normalized family of 2d−1 Hermitian operators depending on a parameter τ. Positivity of the latter family is the main technical problem. We show that the noncentral operators reduce to three-dimensional invariant blocks, while the remaining central operator is exactly solvable in a two-dimensional subspace. Two unitarily equivalent 3×3 matrix representations describe the same active spectrum, but yield different Gershgorin enclosures because Gershgorin bounds are basis dependent. Combining the stronger endpoint from each representation gives a fully analytic asymmetric interval for τ that guarantees a valid POVM. Combining this POVM with the d−1 MUMs yields an explicit GEAM. Finally, using a maximally entangled realization and the local-hidden-state bound previously derived for generalized equiangular measurements, we obtain an explicit steering inequality and a sufficient criterion guaranteeing quantum violation in every prime dimension d≥5.
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BibTeX
@misc{rutkowski2026analytic,
title = {{Analytic construction of dimension-dependent steering witnesses from generalized equiangular measurements}},
author = {Adam Rutkowski and Katarzyna Siudzińska},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07314},
eprint = {2610.07314},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07314},
}
RIS
TY - UNPB
TI - Analytic construction of dimension-dependent steering witnesses from generalized equiangular measurements
AU - Adam Rutkowski
AU - Katarzyna Siudzińska
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07314
UR - https://arxiv.org/abs/2610.07314
AB - We construct a dimension-dependent family of generalized equiangular measurements that gives explicit steering witnesses in every prime dimension d≥5. The construction consists of d−1 mutually unbiased measurements and an additional normalized family of 2d−1 Hermitian operators depending on a parameter τ. Positivity of the latter family is the main technical problem. We show that the noncentral operators reduce to three-dimensional invariant blocks, while the remaining central operator is exactly solvable in a two-dimensional subspace. Two unitarily equivalent 3×3 matrix representations describe the same active spectrum, but yield different Gershgorin enclosures because Gershgorin bounds are basis dependent. Combining the stronger endpoint from each representation gives a fully analytic asymmetric interval for τ that guarantees a valid POVM. Combining this POVM with the d−1 MUMs yields an explicit GEAM. Finally, using a maximally entangled realization and the local-hidden-state bound previously derived for generalized equiangular measurements, we obtain an explicit steering inequality and a sufficient criterion guaranteeing quantum violation in every prime dimension d≥5.
ER -