Euclidean Distance Matrices and Separations in Communication Complexity Theory

Abstract

A Euclidean distance matrix \(D(\alpha )\) is defined by \(D_{ij}=(\alpha _i-\alpha _j)^2\), where \(\alpha =(\alpha _1,\ldots ,\alpha _n)\) is a real vector. We prove that \(D(\alpha )\) cannot be written as a sum of \(\left[ 2\sqrt{n}-2\right] \) nonnegative rank-one matrices, provided that the coordinates of \(\alpha \) are algebraically independent. As a corollary, we provide an asymptotically optimal separation between the complexities of quantum and classical communication protocols computing a given matrix in expectation.