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Complete sets of mutually unbiased bases are only known to exist inprime-power dimensions. We will describe a few approaches to the problemproving the (non)-existence of four mutually unbiased bases in dimension 6.These will include the notions of Grassmannian distance, quadratic matrixprogramming, semidefinite relaxations to polynomial programming, as well asvarious tools from algebraic geometry.

Some attempts at proving the non-existence of a full set of mutually unbiased bases in dimension 6