Zauner-symmetric Weyl–Heisenberg SIC fiducials

Problem

Does every finite dimension admit a Weyl–Heisenberg SIC fiducial that is an eigenvector of a Zauner Clifford unitary? For \(d\geq2\), define the displacement operators \(D_{\mathbf p}:=D_{p,q}:=X_d^pZ_d^q\) for \(\mathbf p=(p,q)^{\mathsf T}\in\mathbb Z_d^2\), using

\begin{equation} X_d\lvert j\rangle=\lvert j+1\!\!\pmod d\rangle, \qquad Z_d\lvert j\rangle=e^{2\pi i j/d}\lvert j\rangle, \qquad \mathbf p=(p,q)^{\mathsf T}\in\mathbb Z_d^2. \tag{1} \end{equation}

Equation (1) fixes the Weyl–Heisenberg orbit up to irrelevant phases. Let \(U_Z\) be a Clifford unitary whose action on displacement operators is

\begin{equation} U_ZD_{\mathbf p}U_Z^\dagger\doteq D_{F_Z\mathbf p}, \qquad F_Z= \begin{pmatrix} 0&-1\\ 1&-1 \end{pmatrix}, \tag{2} \end{equation}

where indices are reduced modulo \(d\) and \(\doteq\) denotes equality up to phase. Equation (2) fixes the distinguished order-three Clifford symmetry. The question is whether, for every \(d\geq2\), there are a unit vector \(\lvert\phi\rangle\) and a phase \(e^{i\theta}\) such that

\begin{equation} U_Z\lvert\phi\rangle=e^{i\theta}\lvert\phi\rangle, \qquad \bigl|\langle\phi\rvert D_{\mathbf p}\lvert\phi\rangle\bigr|^2 =\frac{1}{d+1} \quad\text{for every }\mathbf p\in\mathbb Z_d^2\setminus\{\mathbf0\}. \tag{3} \end{equation}

Equation (3) simultaneously imposes Zauner symmetry and the SIC overlap equations.

Source

Appleby states the all-dimensional existence of a Zauner-symmetric Weyl–Heisenberg SIC fiducial as Conjecture B [App05].

Progress

Reports do not certify correctness or automatically change the problem's status. Progress policy.

Appleby stated Eq. (3) as Conjecture B and verified it for the numerical fiducials then known, through \(d=45\) [App05].
Larger exact and numerical data sets continue to exhibit the Zauner symmetry, including Weyl–Heisenberg SIC solutions in every dimension through \(d=181\). Finite computations do not prove the universal quantifier in Eq. (3) [ABFG19].
In \(2025\), Appleby, Flammia, and Kopp gave a construction of Zauner-symmetric SICs for every \(d>3\), conditional on the order-one abelian Stark conjecture for real quadratic fields and a special-value identity for the Shintani–Faddeev modular cocycle [AFK25].
In September \(2026\), Radchenko and Wheeler proved algebraicity of real-quadratic Stark–Shintani invariants and the rank-one twisted-convolution identity (Theorems 1 and 7) [RW26]. Appleby, Flammia, and Kopp extended the identity to all admissible tuples and constructed Weyl–Heisenberg-covariant ghost SICs for every \(d>3\) (Theorems 1.2–1.3) [AFK26]. This supplies the special-value identity in the earlier conditional construction; converting the ghost SICs to Hermitian SIC projectors requires the appropriate Galois conjugation.
The manuscript dated 4 October \(2026\), prepared and edited by Danylo Radchenko and Campbell Wheeler, reports Stark reciprocity and unitary Galois conjugates (Theorem 3 and Proposition 7). Section 9.2 claims the existence part of Zauner’s conjecture, providing proposed arithmetic input for the Zauner-symmetric construction. [SU26]. The manuscript has not yet passed peer review. This is recorded as significant reported progress; the catalog status remains Unsolved.

Comment

This record requires both the SIC overlap equations and the specified Zauner Clifford symmetry in Eq. (3) for every \(d\ge2\). The catalog retains Unsolved while documenting the reported arithmetic progress; a status change requires a separate maintainer decision. The symmetry requirement is stronger than Weyl–Heisenberg-covariant SIC existence in every dimension.

References

[App05]: D. M. Appleby, “SIC-POVMs and the Extended Clifford Group,” Journal of Mathematical Physics 46, 052107 (2005).DOI arXiv
[ABFG19]: M. Appleby, I. Bengtsson, S. Flammia, and D. Goyeneche, “Tight Frames, Hadamard Matrices and Zauner’s Conjecture,” Journal of Physics A: Mathematical and Theoretical 52, 295301 (2019).DOI arXiv
[AFK25]: M. Appleby, S. T. Flammia, and G. S. Kopp, “A Constructive Approach to Zauner’s Conjecture via the Stark Conjectures,” preprint, arXiv:2501.03970v2 (2025).DOI arXiv
[RW26]: D. Radchenko and C. Wheeler, “Real Quadratic Fields and Finite Quantum Dilogarithms I,” preprint, arXiv:2609.21892v2 (2026).DOI arXiv
[AFK26]: M. Appleby, S. T. Flammia, and G. S. Kopp, “The Twisted Convolution Identity and Ghost \(r\)-SICs from Finite Quantum Dilogarithms,” preprint, arXiv:2609.39192v1 (2026).arXiv
[SU26]: “Stark Units for Real Quadratic Fields and Reciprocity Laws,” manuscript dated 4 October 2026. Cited by title, date, and URL as requested in the manuscript. Manuscript PDF.link

Zauner-symmetric Weyl–Heisenberg SIC fiducials

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