Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1007.3969 · Journal of Physics A Mathematical and Theoretical

Affine constellations without mutually unbiased counterparts

Stefan Weigert, Thomas Durt

Published Sep 9, 2010• 5 citations
arXivHTMLPDF

Abstract

It has been conjectured that a complete set of mutually unbiased bases in a space of dimension d exists if and only if there is an affine plane of order d . We introduce affine constellations and compare their existence properties with those of mutually unbiased constellations. The observed discrepancies make a deeper relation between the two existence problems unlikely.

DOI
10.1088/1751-8113/43/40/402002

Related papers

MUBs, Polytopes, and Finite Geometries

Ingemar Bengtsson

A complete set of N + 1 mutually unbiased bases (MUBs) exists in Hilbert spaces of dimension N = pk, where p is a prime number. They mesh naturally with finite affine planes of order N, that exist when N = pk. The existence of MUBs for other values of N is an open question, and the same is true for finite affine planes. I explore the question whether the existence of complete sets of MUBs is directly related to the existence of finite affine planes. Both questions can be shown to be geometrical questions about a convex polytope, but not in any obvious way the same question.

arXiv:quant-ph/0406174•AIP conference proceedings•Jan 1, 2005•15 citations

Mutually unbiased bases and finite projective planes

Метод Санига, Michel R. P. Planat, Haret Codratian Rosu

It is conjectured that the question of the existence of projective planes whose order is not a power of prime is intimately linked with the problem whether there exists a set of d+1 mutually unbiased bases in a d-dimensional Hilbert space if d differs from a power of prime.

arXiv:math-ph/0403057•Journal of Optics B Quantum and Semiclassical Optics•Jul 24, 2004•93 citations

MUBS, POLYTOPES, AND FINITE GEOMETRIES 1

Ingemar Bengtsson

A complete set of N+1 mutually unbiased bases (MUBs) exists in Hilbert spaces of dimension N = p^k, where p is a prime number. They mesh naturally with finite affine planes of order N, that exist when N = p^k. The existence of MUBs for other values of N is an open question, and the same is true for finite affine planes. I explore the question whether the existence of complete sets of MUBs is directly related to the existence of finite affine planes. Both questions can be shown to be geometrical questions about a convex polytope, but not in any obvious way the same question.

•Jan 1, 2004•25 citations

Maximal sets of mutually unbiased quantum states in dimension 6

STEPHEN D. BRIERLEY, Stefan Weigert

We study sets of pure states in a Hilbert space of dimension $d$ which are mutually unbiased (MU), that is, the moduli of their scalar products are equal to zero, one, or $1∕\sqrt{d}$. Each of these sets will be called a MU constellation, and if four MU bases were to exist for $d=6$, they would give rise to 35 different MU constellations. Using a numerical minimization procedure, we are able to identify only 18 of them in spite of extensive searches. The missing MU constellations provide the strongest numerical evidence so far that no seven MU bases exist in dimension 6.

arXiv:0808.1614•Physical Review A•Oct 13, 2008•92 citations

Mutually Unbiased Bases in Composite Dimensions – A Review

Daniel McNulty, Stefan Weigert

Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.

arXiv:2410.23997•Quantum•Apr 1, 2026•8 citations

Mutually Unbiased Bases and Orthogonal Latin Squares -- version 3

Stefan Joka

In this paper, we prove that the existence of a complete set of mutually unbiased bases (MUBs) in N-dimensional Hilbert space implies the existence of a complete set of mutually orthogonal Latin squares (MOLSs) of order N. In particular, we prove that a complete set of MUBs does not exist in dimension six (the first dimension which is not a power of prime).

quant-pharXiv:2511.03537•Nov 6, 2025•0 citations
It has been conjectured that a complete set of mutually unbiased bases in a space of dimension d exists if and only if there is an affine plane of order d . We introduce affine constellations and compare their existence properties with those of mutually unbiased constellations. The observed discrepancies make a deeper relation between the two existence problems unlikely.
A complete set of N + 1 mutually unbiased bases (MUBs) exists in Hilbert spaces of dimension N = pk, where p is a prime number. They mesh naturally with finite affine planes of order N, that exist when N = pk. The existence of MUBs for other values of N is an open question, and the same is true for finite affine planes. I explore the question whether the existence of complete sets of MUBs is directly related to the existence of finite affine planes. Both questions can be shown to be geometrical questions about a convex polytope, but not in any obvious way the same question.
A complete set of N+1 mutually unbiased bases (MUBs) exists in Hilbert spaces of dimension N = p^k, where p is a prime number. They mesh naturally with finite affine planes of order N, that exist when N = p^k. The existence of MUBs for other values of N is an open question, and the same is true for finite affine planes. I explore the question whether the existence of complete sets of MUBs is directly related to the existence of finite affine planes. Both questions can be shown to be geometrical questions about a convex polytope, but not in any obvious way the same question.
We study sets of pure states in a Hilbert space of dimension ddd which are mutually unbiased (MU), that is, the moduli of their scalar products are equal to zero, one, or 1∕d1∕\sqrt{d}1∕d​. Each of these sets will be called a MU constellation, and if four MU bases were to exist for d=6d=6d=6, they would give rise to 35 different MU constellations. Using a numerical minimization procedure, we are able to identify only 18 of them in spite of extensive searches. The missing MU constellations provide the strongest numerical evidence so far that no seven MU bases exist in dimension 6.
Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.