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:quant-ph/0406174 · AIP conference proceedings

MUBs, Polytopes, and Finite Geometries

Ingemar Bengtsson

Published Jan 1, 2005• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.1874558

Related papers

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

Mutually Unbiased Bases and the Complementarity Polytope

Ingemar Bengtsson, Åsa Ericsson

A complete set of N + 1 mutually unbiased bases (MUBs) forms a convex polytope in the N2 - 1 dimensional space of N × N Hermitian matrices of unit trace. As a geometrical object such a polytope exists for all values of N, while it is unknown whether it can be made to lie within the body of density matrices unless N = pk, where p is prime. We investigate the polytope in order to see if some values of N are geometrically singled out. One such feature is found: It is possible to select N2 facets in such a way that their centers form a regular simplex if and only if there exists an affine plane of order N. Affine planes of order N are known to exist if N = pk; perhaps they do not exist otherwise. However, the link to the existence of MUBs — if any — remains to be found.

arXiv:quant-ph/0410120•Open Systems & Information Dynamics•May 31, 2005•53 citations

MUBs inequivalence and affine planes

William M. Kantor

There are fairly large families of unitarily inequivalent complete sets of N + 1 mutually unbiased bases (MUBs) known in $\mathbb{C}$CN for various prime powers N. The number of such sets is not bounded above by any polynomial as a function of N. While it is standard that there is a superficial similarity between complete sets of MUBs and finite affine planes, there is an intimate relationship between these large families and affine planes. This note briefly summarizes “old” results that do not appear to be well known concerning known families of complete sets of MUBs and their associated planes.

arXiv:1104.3370•Journal of Mathematical Physics•Mar 1, 2012•38 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

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: From finite projective geometry to quantum phase enciphering

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

This short note highlights the most prominent mathematical problems and physical questions associated with the existence of the maximum sets of mutually unbiased bases (MUBs) in the Hilbert space of a given dimension

•Sep 16, 2004•4 citations
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.
A complete set of N + 1 mutually unbiased bases (MUBs) forms a convex polytope in the N2 - 1 dimensional space of N × N Hermitian matrices of unit trace. As a geometrical object such a polytope exists for all values of N, while it is unknown whether it can be made to lie within the body of density matrices unless N = pk, where p is prime. We investigate the polytope in order to see if some values of N are geometrically singled out. One such feature is found: It is possible to select N2 facets in such a way that their centers form a regular simplex if and only if there exists an affine plane of order N. Affine planes of order N are known to exist if N = pk; perhaps they do not exist otherwise. However, the link to the existence of MUBs — if any — remains to be found.
There are fairly large families of unitarily inequivalent complete sets of N + 1 mutually unbiased bases (MUBs) known in C\mathbb{C}CCN for various prime powers N. The number of such sets is not bounded above by any polynomial as a function of N. While it is standard that there is a superficial similarity between complete sets of MUBs and finite affine planes, there is an intimate relationship between these large families and affine planes. This note briefly summarizes “old” results that do not appear to be well known concerning known families of complete sets of MUBs and their associated planes.