A real symmetric matrix M is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size d. The smallest such d is called the (complex) completely positive semidefinite rank of M, and it is an open question whether there exists an upper bound on this number as a function of the matrix size. We construct completely positive semidefinite matrices of size 4k2+2k+2 with complex completely positive semidefinite rank 2k for any positive integer k. This shows that if such an upper bound exists, it has to be at least exponential in the matrix size. For this we exploit connections to quantum information theory and we construct extremal bipartite correlation matrices of large rank. We also exhibit a class of completely positive matrices with quadratic (in terms of the matrix size) completely positive rank, but with linear completely positive semidefinite rank, and we make a connection to the existence of Hadamard matrices.

Linear Algebra and Its Applications
Networks and Optimization

Gribling, S., de Laat, D., & Laurent, M. (2017). Matrices with high completely positive semidefinite rank. Linear Algebra and its Applications, 513, 122–148. doi:10.1016/j.laa.2016.10.015