bibkey: tabiaappleby2013qutrit authors: Gelo Noel M. Tabia; D. M. Appleby year: 2013 title: “Exploring the geometry of qutrit state space using symmetric informationally complete probabilities” doi: 10.1103/PhysRevA.88.012131 url: https://arxiv.org/abs/1304.8075v2 claim: “Every Weyl-Heisenberg SIC in d=3 is obtained by an extended Clifford (anti)unitary from a SIC with fiducial vector (0, 1, -e^{2it})/sqrt(2), t in [0, pi/6], and distinct t in that range give distinct extended Clifford orbits.” strata_touched:

• D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation license: citation-only triage: anchor

Tabia and Appleby, the continuous family of qutrit SIC fiducials

G. N. M. Tabia and D. M. Appleby, Exploring the geometry of qutrit state space using symmetric informationally complete probabilities, Phys. Rev. A 88, 012131 (2013); arXiv:1304.8075v2 (quant-ph).

Verified locator

DOI: 10.1103/PhysRevA.88.012131 (Crossref: Physical Review A, volume 88, article 012131, issued 2013-07-31). Preprint: https://arxiv.org/abs/1304.8075v2. The arXiv abstract page lists versions v1 and v2; the TeX source qutritGeom.tex of the v2 e-print supplies the statements below.

Source statements

Section II, “Properties of Weyl-Heisenberg qutrit SICs”, after the clock and shift matrices of $d=3$:

Every Weyl-Heisenberg SIC in $d=3$ can be obtained by acting with an (extended) Clifford (anti)unitary on a SIC with fiducial vector $|{\psi }_t⟩=\frac{1}{\sqrt{2}}(0,1,-e^{2it}{)}^{\mathsf{T}}$, $t\in [0,\frac{\pi }{6}]$. Fiducials corresponding to distinct values of $t$ in the range $[0,\frac{\pi }{6}]$ generate distinct orbits of the extended Clifford group.

The paper attributes the orbit classification to Appleby (2005): three types of orbits contain $|{\psi }_t⟩$, the infinitely many generic ones for $t\in (0,\frac{\pi }{6})$ and two exceptional ones at the endpoints. The abstract states that the infinitely many qutrit SICs fall into eight SIC families corresponding to independent orbits of the extended Clifford group.

Use in this repository

D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation uses the vectors ${\psi }_z=(0,1,-z)/\sqrt{2}$, $|z|=1$. For $z=e^{2it}$ with $t\in [0,\pi /6]$ they are the fiducials quoted above. The module proves the SIC overlap condition for every unit $z$ directly (qutritFiducial_isSIC), so it does not rely on the paper for the remaining values of $z$; only the family itself is attributed to this source.