# Euclidean Distance Matrices and Separations in Communication Complexity Theory

Article

## 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.

## Keywords

Nonnegative matrix factorization Extended formulations of polytopes Positive semidefinite rank Communication complexity

## Mathematics Subject Classification

15A23 52B12 81P45

## Notes

### Acknowledgements

I would like to thank Troy Lee for pointing my attention to this problem and helpful comments. I am also grateful to the editor János Pach for handling my paper and to anonymous referees for helpful comments and suggestions, which allowed me to improve the presentation of my results. The article was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2017–2018 (Grant No. 17-01-0049) and by the ‘Russian Academic Excellence Project 5-100’.

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