论文标题
互明确函数方法的双重界限
Dual bounds for the positive definite functions approach to mutually unbiased bases
论文作者
论文摘要
一个长期存在的公开问题询问是否可以在$ \ mathbb {c}^6 $中存在7个相互公正的基础（MUB），或者更一般而言，对于任何$ \ Mathbb {C}^d $，对于任何$ d $而言，$ d + 1 $ mubs in $ \ mathbb {c}^d $都不是主要的力量。 Kolountzakis，Matolcsi和Weiner（2016）最近的工作提出了一种正面确定功能的方法（Delsarte方法在编码理论中的相对，以及Lovász的半际计划放宽独立设置问题的放宽）作为负面回答这个问题的手段。也就是说，他们询问是否存在满足各种属性的单位矩阵输入的多项式，通过正面确定函数的方法，这将表明7个Mubs在$ \ mathbb {c}^6 $中不存在。使用凸双重参数，我们证明了这种程度最多的6个多项式是不存在的。我们还提出了一份通用双重证书，我们可以猜测，以证明此方法绝不能够证明，严格存在$ \ \ Mathbb {c}^d $中的$ d + 1 $ mubs。
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 Lovász'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 $\mathbb{C}^d$.
