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arXiv:1009.2407 · Studia Scientiarum Mathematicarum Hungarica

A fourier analytic approach to the problem of mutually unbiased bases

Máté Matolcsi

Published Nov 17, 2012• 8 citations
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Abstract

We give a new approach to the problem of mutually unbiased bases (MUBs), based on a Fourier analytic technique borrowed from additive combinatorics. The method provides a short and elegant generalization of the fact that there are at mostd+ 1 MUBs in ℂd. It may also yield a proof that no complete system of MUBs exists in some composite dimensions — a long standing open problem.

DOI
10.1556/sscmath.49.2012.4.1221

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We give a new approach to the problem of mutually unbiased bases (MUBs), based on a Fourier analytic technique borrowed from additive combinatorics. The method provides a short and elegant generalization of the fact that there are at mostd+ 1 MUBs in ℂd. It may also yield a proof that no complete system of MUBs exists in some composite dimensions — a long standing open problem.
We present a new approach to the problem of mutually unbiased bases (MUBs), based on positive definite functions on the unitary group. The method provides a new proof of the fact that there are at most d+1d+1d+1 MUBs in Cd{\mathbb C}^dCd. It may also lead to a proof of non-existence of complete systems of MUBs in dimension 6 via a conjectured algebraic identity.
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.
A collection of pairwise mutually unbiased bases (in short: MUB) in d>1 dimensions may consist of at most d+1 bases. Such "complete" collections are known to exists in C^d when d is a power of a prime. However, in general little is known about the maximal number N(d) of bases that a collection of MUBs in C^d can have. In this work it is proved that a collection of d MUBs in C^d can be always completed. Hence N(d) cannot be d and when d>1 we have a dichotomy: either N(d)=d+1 (so that there exists a complete collection of MUBs), or N(d)\leq d-1. In the course of the proof an interesting new characterization is given for a linear subspace of M_d(C) to be a subalgebra.
A collection of pairwise mutually unbiased bases (in short: MUB) in d > 1 d>1 dimensions may consist of at most d + 1 d+1 bases. Such “complete” collections are known to exist in C d \mathbb {C}^d when d d is a power of a prime. However, in general, little is known about the maximum number N ( d ) N(d) of bases that a collection of MUB in C d \mathbb {C}^d can have. In this work it is proved that a collection of d d MUB in C d \mathbb {C}^d can always be completed. Hence N ( d ) ≠ d N(d)\neq d , and when d > 1 d>1 we have a dichotomy: either N ( d ) = d + 1 N(d)=d+1 (so that there exists a complete collection of MUB) or N ( d ) ≤ d − 1 N(d)\leq d-1 . In the course of the proof an interesting new characterization is given for a linear subspace of