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MUBs: From finite projective geometry to quantum phase enciphering

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

Published Sep 16, 2004• 4 citations
OpenAlex

Abstract

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

Related papers

MUBs: From Finite Projective Geometry to Quantum Phase Enciphering

H. C. Rosu

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.

arXiv:quant-ph/0409096•AIP conference proceedings•Jan 1, 2004•5 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

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

On the mathematical foundations of mutually unbiased bases

Koen Thas

In order to describe the right setting to handle Zauner's conjecture on mutually unbiased bases (MUBs) (saying that in $\mathbb{C}^d$, a set of MUBs of the theoretical maximal size $d + 1$ exists only if $d$ is a prime power), we pose some fundamental questions which naturally arise. Some of these questions have important consequences for the construction theory of (new) sets of maximal MUBs.

arXiv:1409.3386•Sep 11, 2014•0 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

Classifying all mutually unbiased bases in Rel

Evans, Julia, Duncan, Ross, Alex Lang +1 more

Finding all the mutually unbiased bases in various dimensions is a problem of fundamental interest in quantum information theory and pure mathematics. The general problem formulated in finite-dimensional Hilbert spaces is open. In the categorical approach to quantum mechanics one can find examples of categories which behave ``like'' the category of finite-dimensional Hilbert spaces in various ways but are subtly different. One such category is the category of sets and relations, $\mathbf{Rel}$. One can formulate the concept of mutually unbiased bases here as well. In this note we classify all the mutually unbiased bases in this category by relating it to a standard question in combinatorics.

arXiv:0909.4453•Sep 24, 2009•11 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.
In order to describe the right setting to handle Zauner's conjecture on mutually unbiased bases (MUBs) (saying that in Cd\mathbb{C}^dCd, a set of MUBs of the theoretical maximal size d+1d + 1d+1 exists only if ddd is a prime power), we pose some fundamental questions which naturally arise. Some of these questions have important consequences for the construction theory of (new) sets of maximal MUBs.
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.
Finding all the mutually unbiased bases in various dimensions is a problem of fundamental interest in quantum information theory and pure mathematics. The general problem formulated in finite-dimensional Hilbert spaces is open. In the categorical approach to quantum mechanics one can find examples of categories which behave ``like'' the category of finite-dimensional Hilbert spaces in various ways but are subtly different. One such category is the category of sets and relations, Rel\mathbf{Rel}Rel. One can formulate the concept of mutually unbiased bases here as well. In this note we classify all the mutually unbiased bases in this category by relating it to a standard question in combinatorics.