Lower bounds for maximal cp-ranks of completely positive matrices and tensors

Abstract

Let $p_n$ denote the maximal cp-rank attained by completely positive $n\times n$ matrices. Only lower and upper bounds for $p_n$ are known, when $n\ge6$, but it is known that $p_n=\frac{n^2}2\big(1+o(1)\big)$, and the difference of the current best upper and lower bounds for $p_n$ is of order $\mathcal{O}\big(n^{3/2}\big)$. In this paper, that gap is reduced to $\mathcal{O}\big(n\log\log n\big)$. To achieve this result, a sequence of generalized ranks of a given matrix A has to be introduced. Properties of that sequence and its generating function are investigated. For suitable A, the $d$th term of that sequence is the cp-rank of some completely positive tensor of order $d$. This allows the derivation of asymptotically matching lower and upper bounds for the maximal cp-rank of completely positive tensors of order $d>2$ as well.