The complexity of positive semidefinite matrix factorization

Let $A$ be a matrix with nonnegative real entries. The PSD rank of $A$ is the smallest integer $k$ for which there exist $k\times k$ real PSD matrices $B1,\ldots,Bm$, $C1,\ldots,Cn$ satisfying $A(i|j)=\operatorname{tr}(BiCj)$ for all $i,j$. This paper determines the computational complexity status of the PSD rank. Namely, we show that the problem of computing this function is polynomial-time equivalent to the existential theory of the reals.