OP-3060OpenQuantum information

Existence of SIC-POVMs

Prove or disprove that for every $d\ge2$ there exist $d^2$ unit vectors $\{|\psi_j\rangle\}\subset\mathbb{C}^d$ with $|\langle\psi_j|\psi_k\rangle|^2=\tfrac{1}{d+1}$ for $j\neq k$ .

Source-reported statusAs of 2026-09-04Source & evidence ↓

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Problem statement

Prove or disprove that for every $d\ge2$ there exist $d^2$ unit vectors $\{|\psi_j\rangle\}\subset\mathbb{C}^d$ with $|\langle\psi_j|\psi_k\rangle|^2=\tfrac{1}{d+1}$ for $j\neq k$ .

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Background

A symmetric informationally complete POVM (SIC) is obtained from $d^2$ equiangular unit vectors in $\mathbb C^d$ , with effects $|\psi_j\rangle\langle\psi_j|/d$ . SICs are minimal informationally complete measurements, useful in tomography and quantum foundations. This is a measurement-geometry problem, not intrinsically a Bell-nonlocality question.

Exact algebraic constructions and extensive high-precision numerical solutions provide strong evidence, but neither establishes all-dimensional existence. The unrestricted existence question is weaker than requiring Weyl–Heisenberg covariance or the additional order-three symmetry associated with Zauner's conjecture. These variants should not be conflated.

Appleby, Flammia, Kopp, Yard and others connected SICs with abelian extensions of real quadratic fields and Hilbert's twelfth problem. Appleby–Flammia–Kopp's constructive approach proves validity under two number-theoretic conjectures: an order-one abelian Stark conjecture and a special-value identity for the Shintani–Faddeev modular cocycle. This is conditional progress, not an unconditional existence theorem.

A general proof or one dimension admitting no SIC resolves the stated sharp question.

Progress & source assessment

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

References

[1]Appleby, Flammia & Kopp, 'A Constructive Approach to Zauner’s Conjecture via the Stark Conjectures' (conditional construction, 2025) link
[2]Renes, Blume-Kohout, Scott, Caves, 'Symmetric informationally complete quantum measurements', J. Math. Phys. 45, 2171 (2004) link
[3]Appleby, Flammia, McConnell, Yard, 'SICs and algebraic number theory', Found. Phys. 47 (2017) link
[4]Scott & Grassl, 'SIC-POVMs: A new computer study', J. Math. Phys. 51 (2010) link

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