Matrices with high completely positive semidefinite rank

TL;DR

This paper constructs specific large matrices with high completely positive semidefinite rank, demonstrating that if an upper bound exists, it must grow exponentially with matrix size, with implications for quantum information theory.

Contribution

It provides explicit constructions of matrices with high completely positive semidefinite rank, establishing lower bounds and exploring connections to quantum information and Hadamard matrices.

Findings

Constructed matrices of size $4k^2+2k+2$ with rank $2^k$

Showed lower bound on rank growth is exponential in size

Connected completely positive semidefinite rank to quantum correlations

Abstract

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 $4k^2+2k+2$ with complex completely positive semidefinite rank $2^k$ 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.