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

Let p _n denote the maximal cp-rank attained by completely positive n × n matrices. Only lower and upper bounds for p _n are known, when n ≥ 6, but it is known that (Formula Presented), and the difference of the current best upper and lower bounds for p _n is of order O(n ^3/2). In this paper, that gap is reduced to O(n log log n). 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 dth 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.