Real Fiducial Vectors Only Exist in Dimensions d≡ 3 pmod4
Abstract
We investigate restrictions on the dimensions and coordinate fields of real Weyl-Heisenberg fiducial vectors. Our main result is that such vectors can exist only in dimensions d≡ 3pmod4. This result is motivated by evidence showing that real Weyl-Heisenberg fiducial vectors can exist only in dimensions d = 4m2 + 3, m ∈ ℤ≥ 0. We further restrict the possible dimensions when the coordinate field is a totally real number field, an even or odd degree extension of ℚ, a real quadratic field, or a maximal real subfield of a cyclotomic field. Lastly, we establish non-existence over odd-degree number fields under either positivity of all non-trivial cyclic-shift overlaps or a conjectural containment of a distinguished quadratic field in the SIC field.
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BibTeX
@misc{loo2026real,
title = {{Real Fiducial Vectors Only Exist in Dimensions $d\equiv 3 \pmod4$}},
author = {Igor Van Loo and Frédérique Oggier},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13168},
eprint = {2610.13168},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13168},
}
RIS
TY - UNPB
TI - Real Fiducial Vectors Only Exist in Dimensions d≡ 3 pmod4
AU - Igor Van Loo
AU - Frédérique Oggier
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13168
UR - https://arxiv.org/abs/2610.13168
AB - We investigate restrictions on the dimensions and coordinate fields of real Weyl-Heisenberg fiducial vectors. Our main result is that such vectors can exist only in dimensions d≡ 3pmod4. This result is motivated by evidence showing that real Weyl-Heisenberg fiducial vectors can exist only in dimensions d = 4m^2 + 3, m ∈ ℤ_≥ 0. We further restrict the possible dimensions when the coordinate field is a totally real number field, an even or odd degree extension of ℚ, a real quadratic field, or a maximal real subfield of a cyclotomic field. Lastly, we establish non-existence over odd-degree number fields under either positivity of all non-trivial cyclic-shift overlaps or a conjectural containment of a distinguished quadratic field in the SIC field.
ER -