Everyone is welcome to attend the next N&O seminar with Sven Polak with the title 'Mutually unbiased bases: polynomial optimization and symmetry'.
Abstract: A set of k orthonormal bases of \C^d is called mutually unbiased if |<e,f>|=1/sqrt(d) whenever e and f are basis vectors in distinct bases. A notorious question is for which pairs (d,k) there exist k mutually unbiased bases in dimension d. The upper bound k<=d+1 is attained when d is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain C*-algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of S_d and S_k.
We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the module \C^{([d] x [k])^t} for the wreath product of S_d and S_k into irreducible modules. We present numerical results for small d,k and low levels of the hierarchy.
This is based on joint work with Sander Gribling (IRIF Paris), see arXiv:2111.05698
