Weyl–Heisenberg-covariant SICs in every dimension

Problem

Does every finite dimension admit a symmetric informationally complete measurement that is a single Weyl–Heisenberg orbit? For an integer \(d\geq2\), let \(\omega_d=e^{2\pi i/d}\) and define shift, phase, and displacement operators on the computational basis by

\begin{equation} X_d\lvert j\rangle=\lvert j+1\!\!\pmod d\rangle, \qquad Z_d\lvert j\rangle=\omega_d^j\lvert j\rangle, \qquad D_{p,q}=X_d^pZ_d^q, \quad (p,q)\in\mathbb Z_d^2. \tag{1} \end{equation}

Equation (1) fixes a phase convention that does not affect the orbit of rank-one projectors. The question is whether, for every \(d\geq2\), there is a unit vector \(\lvert\phi\rangle\in\mathbb C^d\) satisfying

\begin{equation} \bigl|\langle\phi\rvert D_{p,q}\lvert\phi\rangle\bigr|^2 =\frac{1}{d+1} \qquad \text{for every }(p,q)\in\mathbb Z_d^2\setminus\{(0,0)\}. \tag{2} \end{equation}

If Eq. (2) holds, the \(d^2\) projectors in the Weyl–Heisenberg orbit of \(\lvert\phi\rangle\) form a SIC.

Source

Renes, Blume-Kohout, Scott, and Caves explicitly conjecture the existence of a Weyl–Heisenberg-covariant SIC in every finite dimension [RBS+04].

Progress

Renes, Blume-Kohout, Scott, and Caves stated Eq. (2) in every dimension as Conjecture 1 and found numerical solutions through \(d=45\) [RBS+04].
Appleby, Bengtsson, Flammia, and Goyeneche reported numerical Weyl–Heisenberg SICs in every dimension through \(d=181\) and in many larger dimensions. These computations establish individual finite cases, not the all-dimensional assertion [ABFG19].
Appleby, Flammia, and Kopp give a construction for all \(d>3\) conditional on the order-one abelian Stark conjecture and a special-value identity for the Shintani–Faddeev modular cocycle. The hypotheses remain unproved, so the construction is not unconditional [AFK25].

Comment

The unresolved step is an unconditional construction, or existence proof, for Eq. (2) in every \(d\). Problem 19 imposes the additional requirement that the fiducial be Zauner symmetric.

References

[RBS+04]: J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric Informationally Complete Quantum Measurements,” Journal of Mathematical Physics 45, 2171–2180 (2004).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

Weyl–Heisenberg-covariant SICs in every dimension

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