Real Fiducial Vectors Only Exist in Dimensions $d\equiv 3 \pmod{4}$

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 $d\equiv 3\pmod{4}$. This result is motivated by evidence showing that real Weyl-Heisenberg fiducial vectors can exist only in dimensions $d = 4m^2 + 3, m \in \mathbb{Z}_{\geq 0}$. We further restrict the possible dimensions when the coordinate field is a totally real number field, an even or odd degree extension of $\mathbb{Q}$, 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.

Number Theory (math.NT); Quantum Physics (quant-ph)

arXiv:2610.13168 [math.NT]

(or arXiv:2610.13168v1 [math.NT] for this version)

https://doi.org/10.48550/arXiv.2610.13168 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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