  Correlation matrices (positive semidefinite matrices with ones on the
diagonal) are of fundamental interest in quantum information theory. In this
work we introduce and study the set of $r$-decomposable correlation matrices:
those that can be written as the Schur product of correlation matrices of rank
at most $r$. We find that for all $r \geq 2$, every $(r+1) \times (r+1)$
correlation matrix is $r$-decomposable, and we construct ${(2r+1) \times
(2r+1)}$ correlation matrices that are not $r$-decomposable. One question this
leaves open is whether every $4 \times 4$ correlation matrix is
$2$-decomposable, which we make partial progress toward resolving. We apply our
results to an entanglement detection scenario.
