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arXiv:quant-ph/0409096 · AIP conference proceedings

MUBs: From Finite Projective Geometry to Quantum Phase Enciphering

H. C. Rosu

Published Jan 1, 2004• 5 citations
arXivHTMLPDF

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.
DOI
10.1063/1.1834443

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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
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.
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.
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.
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.