Zauner's Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $d\geq1$, equip $\mathbb C^d$ with its usual Hermitian inner product $\langle u,v\rangle=\sum_{j=1}^d\overline{u_j}v_j$. A symmetric informationally complete positive operator-valued measure (SIC-POVM) is a family $E_1,\ldots,E_{d^2}$ of positive rank-one operators such that $\sum_iE_i=I_d$ and \[ E_i=\frac1d\psi_i\psi_i^*,\qquad \|\psi_i\|=1,\qquad |\langle\psi_i,\psi_j\rangle|^2=\frac1{d+1}\quad(i\ne j). \] The operators $\psi_i\psi_i^*$ are rank-one orthogonal projectors, and the displayed conditions are equivalent to $d^2$ equiangular complex lines (with the trivial $d=1$ case interpreted separately).

Statement recorded in the supplied source.

For every positive integer $d$, do there exist unit vectors $\psi_1,\ldots,\psi_{d^2}\in\mathbb C^d$ such that \[ |\langle\psi_i,\psi_j\rangle|^2=\frac1{d+1}\quad(i\ne j),\qquad \sum_{i=1}^{d^2}\psi_i\psi_i^*=dI_d? \] This asks for SIC existence in every finite dimension, with no additional Weyl--Heisenberg covariance or prescribed order-three symmetry requirement. [S2]

Short English statement

Does every finite-dimensional complex Hilbert space admit a symmetric informationally complete quantum measurement?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1106 (ALG-006). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Marcus Appleby, Steven T. Flammia and Gene S. Kopp, A Constructive Approach to Zauner’s Conjecture via the Stark Conjectures, arXiv:2501.03970v2 (2025), §1, Definitions 1.1--1.2 and Conjecture 1.3. The separate symmetry conjectures and conditional construction are distinguished from universal SIC existence. https://arxiv.org/abs/2501.03970.