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A gap for the maximum number of mutually unbiased bases

Mihály Weiner

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 ofA 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

Proceedings of the American Mathematical SocietyJan 23, 201335 citations

On the mathematical foundations of mutually unbiased bases

Koen Thas

Quantum Information ProcessingDec 21, 20173 citations

A gap for the maximum number of mutually unbiased bases

Mihály Weiner

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 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.

arXiv:0902.0635Feb 4, 20094 citations

The problem of mutually unbiased bases in dimension 6

Philippe Jaming, Máté Matolcsi, Péter Móra

Cryptography and CommunicationsApr 12, 201019 citations

A fourier analytic approach to the problem of mutually unbiased bases

Máté Matolcsi

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.

arXiv:1009.2407Studia Scientiarum Mathematicarum HungaricaNov 17, 20128 citations

The problem of mutually unbiased bases in dimension 6

JamingPhilippe, MatolcsiMáté, MóraPéter

Cryptography and CommunicationsSep 1, 20100 citations

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