SIC-POVM existence in every dimension

Problem

Does a symmetric informationally complete positive-operator-valued measure exist in every finite dimension \(d\ge2\)? Equivalently, determine whether for every such \(d\) there are \(d^2\) unit vectors \(\lvert\psi_1\rangle,\ldots,\lvert\psi_{d^2}\rangle\in\mathbb{C}^d\) satisfying

\begin{equation} \sum_{j=1}^{d^2} \lvert\psi_j\rangle\!\langle\psi_j\rvert=dI_d, \qquad \left|\langle\psi_j\vert\psi_k\rangle\right|^2 =\frac{1}{d+1} \quad\text{for all }j\ne k. \tag{1} \end{equation}

When Eq. (1) holds, the effects \(\Pi_j=d^{-1}\lvert\psi_j\rangle\!\langle\psi_j\rvert\) form the desired SIC-POVM. No covariance or additional symmetry is required.

Source

Renes, Blume-Kohout, Scott, and Caves explicitly conjecture that SIC-POVMs exist in every finite dimension [RBS+04].

Progress

Renes, Blume-Kohout, Scott, and Caves explicitly conjectured existence in every dimension and found Weyl–Heisenberg-covariant numerical solutions through \(d=45\). Numerical solutions do not prove the universal quantifier in Eq. (1) [RBS+04].
Horodecki, Rudnicki, and Życzkowski identify construction of SIC-POVMs in an unbounded sequence of finite dimensions as an intermediate open milestone. That milestone is implied by the universal statement in Eq. (1), but it remains unproved unconditionally [HRZ22].
Appleby, Bengtsson, Flammia, and Goyeneche reported numerical solutions in every dimension through \(d=181\) and in many larger dimensions. The computations provide extensive evidence but no all-dimensional construction [ABFG19].
Appleby, Flammia, and Kopp constructed SICs in every dimension \(d>3\) conditional on two unproved number-theoretic conjectures. The conditional hypotheses prevent this result from establishing Eq. (1) unconditionally [AFK25].
A claimed unconditional proof posted in \(2026\) was withdrawn after the author stated that the proof was incorrect, so it does not change the status of the problem [Jok26].

Comment

The remaining gap is unconditional existence for every integer \(d\ge2\) in Eq. (1). Numerical evidence, covariance-restricted constructions, and results conditional on number-theoretic conjectures do not settle the unrestricted existence question. The weaker objective of an unbounded sequence of finite dimensions is subsumed here as an intermediate milestone rather than maintained as a separate problem.

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
[HRZ22]: P. Horodecki, Ł. Rudnicki, and K. Życzkowski, “Five Open Problems in Quantum Information Theory,” PRX Quantum 3, 010101 (2022).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).DOI arXiv
[Jok26]: S. Joka, “Symmetric Informationally Complete Positive Operator Valued Measure and Zauner Conjecture,” withdrawn preprint, arXiv:2601.13475v5 (2026).arXiv

SIC-POVM existence in every dimension

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