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 {C}^d$$ for any d that is not a prime power. Recent work of Proc Am Math Soc 146(3), 1143–1150 (2018) proposed an application the method positive definite functions (a relative Delsarte’s coding theory and Lovász’s semidefinite programming relaxation independent set problem) as means answering this question negative. Namely, they ask whether exists polynomial unitary matrix input satisfying various properties which, through functions, would show cannot . Using convex duality argument, we prove no such degree at most 6. We also propose general dual certificate which conjecture to certify never strictly fewer than