Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\mathbb{C}^d$.

10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
18
The categories of this paper are: econ
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas