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arXiv:quant-ph/0610216 · AIP conference proceedings

Three Ways to Look at Mutually Unbiased Bases

Ingemar Bengtsson

Published Jan 1, 2007• 118 citations
arXivHTMLPDF

Abstract

This is a review of the problem of Mutually Unbiased Bases in finite dimensional Hilbert spaces, real and complex. Also a geometric measure of “mubness” is introduced, and applied to some explicit calculations in six dimensions (partly done by Björck and by Grassl). Although this does not yet solve any problem, some appealing structures emerge.

DOI
10.1063/1.2713445

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This is a review of the problem of Mutually Unbiased Bases in finite dimensional Hilbert spaces, real and complex. Also a geometric measure of “mubness” is introduced, and applied to some explicit calculations in six dimensions (partly done by Björck and by Grassl). Although this does not yet solve any problem, some appealing structures emerge.
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.
We present and generalize the basic ideas underlying recent work aimed at the construction of mutually unbiased bases in finite dimensional Hilbert spaces with the help of group and graph theoretical concepts. In this approach finite groups are used to construct maximal sets of mutually unbiased bases. Thus the prime number restrictions of previous approaches are circumvented and this construction principle sheds new light onto the intricate relation between mutually unbiased bases and characteristic geometrical structures of Hilbert spaces.
Two orthogonal bases A and B of an n-dimensional Hilbert space are called Mutually Unbiased if for ai ∈ A and bj ∈ B holds: |(ai, bj)| = 1 n for each i, j ∈ {1, . . . , n}. A set B of Mutually Unbiased Bases (MUBs) is a set of bases such that each two distinct bases Bi, Bj ∈ B are mutually unbiased. The bounds on the number of MUBs, dependent on the dimension n, will be derived. Also a construction for MUBs in Cp, with p prime, will be showed. Finally, some open problems will be discussed.
A complex Hilbert space of dimension six supports at least three but not more than seven mutually unbiased bases. Two computer-aided analytical methods to tighten these bounds are reviewed, based on a discretization of parameter space and on Gröbner bases. A third algorithmic approach is presented: the non-existence of more than three mutually unbiased bases in composite dimensions can be decided by a global optimization method known as semidefinite programming. The method is used to confirm that the spectral matrix cannot be part of a complete set of seven mutually unbiased bases in dimension six.