Euclidean distance matrices and separations in communication complexity theory

TL;DR

This paper investigates the structure of Euclidean distance matrices and demonstrates a fundamental limitation on their decomposition, leading to significant implications for the separation of quantum and classical communication complexities.

Contribution

It establishes a lower bound on the nonnegative rank decomposition of Euclidean distance matrices with algebraically independent coordinates, resolving open problems in computation theory.

Findings

Proves Euclidean distance matrices cannot be decomposed into fewer than approximately 2√n nonnegative rank-one matrices.

Provides an asymptotically optimal separation between quantum and classical communication complexities.

Enables solving several open problems in computation theory related to matrix decompositions.

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. 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.