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

A long-standing open problem asks if there can exist 7 mutually unbiased bases (MUBs) in $\mathbb{C}^6$, or, more generally, $d + 1$ MUBs in $\mathbb{C}^d$ for any $d$ that is not a prime power. The recent work of Kolountzakis, Matolcsi, and Weiner (2016) proposed an application of the method of positive definite functions (a relative of Delsarte's method in coding theory and Lov\'{a}sz's semidefinite programming relaxation of the independent set problem) as a means of answering this question in the negative. Namely, they ask whether there exists a polynomial of a unitary matrix input satisfying various properties which, through the method of positive definite functions, would show the non-existence of 7 MUBs in $\mathbb{C}^6$. Using a convex duality argument, we prove that such a polynomial of degree at most 6 cannot exist. We also propose a general dual certificate which we conjecture to certify that this method can never show that there exist strictly fewer than $d + 1$ MUBs in $\mathbb{C}^d$.