Leírás
Abstract:
Two orthonormal bases in \mathbb{C}^d are mutually unbiased if the pairwise inner products have magnitude 1/\sqrt{d}. Let M(d) denote the maximum number of mutually unbiased bases (MUBs) in dimension d.
It is not hard to show that M(d) <= d+1, i.e. that there are at most d+1 mutually unbiased bases (MUBs) in dimension d.
It is also known that if d is a product of prime powers p_1^m_1,...,p_r^m_r for some primes p_1<...<p_r, then M(d)>=p_1^m_1 + 1, i.e. that there exist p_1^m_1+1 MUBs in dimension d.
The smallest d where these estimates leave a gap is for d=6: 3 <= M(6) <= 7.
It was conjectured by Zauner in 1999 that M(6)=3: there exist 3 MUBs (MUB-triplets) but not 4 MUBs in dimension 6. This conjecture remains unsolved; in fact, neither the upper or lower bound of 3<=M(6)<=7 has been improved. I will talk about some conjectures related to the structure of MUB-triplets as well as some partial results related to this problem.
This is a report on work in progress joint with Máté Matolcsi, Dániel Varga and Mihály Weiner.
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