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

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].
Appleby, Flammia, and Kopp construct Zauner-symmetric SICs for all \(d>3\) under two number-theoretic conjectures. Those unproved assumptions leave the unconditional problem open [AFK25].

Comment

The open problem is the unconditional all-dimensional existence of a fiducial satisfying both conditions in Eq. (3). It is strictly stronger than Weyl–Heisenberg-covariant SIC existence in Problem 18.

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,” arXiv:2501.03970 (2025).arXiv

Zauner-symmetric Weyl–Heisenberg SIC fiducials

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