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

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

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].
In their \(2022\) survey, Horodecki, Rudnicki, and Życzkowski identified construction of SIC-POVMs in an unbounded sequence of finite dimensions as an intermediate open milestone. The universal statement in Eq. (1) implies that milestone [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].
In \(2025\), Appleby, Flammia, and Kopp gave a construction of 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].
A claimed unconditional proof posted in \(2026\) was withdrawn after the author stated that the proof was incorrect, so it does not change the status of the problem [Jok26].
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 SIC existence in every dimension by combining these results with the earlier construction. [SU26]. The manuscript has not yet passed peer review. This is recorded as significant reported progress; the catalog status remains Unsolved.

Comment

The catalog retains Unsolved while documenting the reported progress above; a status change requires a separate maintainer decision. The target is unconditional existence for every integer \(d\ge2\) in Eq. (1). 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,” preprint, arXiv:2501.03970v2 (2025).DOI arXiv
[Jok26]: S. Joka, “Symmetric Informationally Complete Positive Operator Valued Measure and Zauner Conjecture,” withdrawn preprint, arXiv:2601.13475v5 (2026).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

SIC-POVM existence in every dimension

Problem

Source

Progress

Comment

References

Cite this problem

BibTeX

Plain text

Problem

Source

Progress

Comment

References

Related problems