Euclidean distance matrices and separations in communication complexity theory

A Euclidean distance matrix $D(\alpha )$ is defined by $D_{ij}=({\alpha }_i-{\alpha }_j{)}^2$, where $\alpha =({\alpha }_1,\dots ,{\alpha }_n)$ is a real vector. We prove that $D(\alpha )$ cannot be written as a sum of $[2\sqrt{n}-2]$ nonnegative rank-one matrices, provided that the coordinates of $\alpha$ are algebraically independent. This result allows one to solve several open problems in computation theory. In particular, we provide an asymptotically optimal separation between the complexities of quantum and classical communication protocols computing a matrix in expectation.