Mutually unbiased bases: polynomial optimization and symmetry

Abstract

A set of k orthonormal bases of C d is called mutually unbiased if | e, f | 2 = 1/d whenever e and f are basis vectors in distinct bases. A natural question is for which pairs (d, k) there exist k mutually unbiased bases in dimension d. The (well-known) 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. Along the way we provide a novel explicit decomposition of a certain "L-shaped" permutation module into irreducible modules. We present numerical results for small d, k and low levels of the hierarchy. In particular, we obtain (numerical) sumof-squares proofs for the (well-known) fact that there do not exist d + 2 mutually unbiased bases in dimensions d = 2, 3, 4, 5.

Three numerical approaches to find mutually unbiased bases using Bell inequalities

“…An alternative approach to the optimisation is to focus on problem (19) a reduction in the size of the search space as well as the memory required. The downside, however, is that the objective function is now nonlinear (not even bi-linear) and thus many efficient solvers (i.e.…”

“…Note that projectivity and self-adjointness implies positive semidefiniteness, and therefore positive semidefiniteness does not need to be imposed. The optimal value of the optimisation problem (19), denoted as…”

“…In the following three sections we apply three numerical methods to solve the optimisation problems (15) and (19) in order to numerically tackle Zauner's conjecture.…”

Dual bounds for the positive definite functions approach to mutually unbiased bases

“…Another line of work, initiated by [24] and continued more recently by [25], uses that computing the size of a MUB may also be cast as an optimization problem over non-commutative (i.e., matrix-valued) variables. Such problems admit a variant of the sum-of-squares hierarchy of convex relaxations, which these works use to formulate different semidefinite programming relaxations.…”

“…Unfortunately, the decomposition of the associated representation into irreducible representations of S n appears to be unknown in general; see, e.g., Chapter 12 of [37]. The recent paper [25], which explores a different approach using non-commutative sum-of-squares optimization to bound the sizes of MUBs, uses this symmetry group as well, albeit only for numerical computations.…”