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

TL;DR

This paper investigates the positive definite functions approach to the MUB problem in complex spaces, proving certain polynomial bounds and proposing a dual certificate conjecture to explain the method's limitations.

Contribution

The authors prove that no polynomial of degree at most 6 can certify the non-existence of 7 MUBs in a6^6 and propose a dual certificate conjecture for the method's fundamental limitations.

Findings

Proved the non-existence of degree a0a0 6 polynomials certifying 7 MUBs in a6^6.

Proposed a dual certificate conjecture suggesting the method cannot show fewer than d+1 MUBs exist.

Provided a convex duality argument to establish polynomial degree bounds.

Abstract

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 $\mathbb{C}^d$.