  The operator Schmidt rank is the minimum number of terms required to express
a state as a sum of elementary tensor factors. Here we provide a new proof of
the fact that any bipartite mixed state with operator Schmidt rank two is
separable, and can be written as a sum of two positive semidefinite matrices
per site. Our proof uses results from the theory of free spectrahedra and
operator systems, and illustrates the use of a connection between
decompositions of quantum states and decompositions of nonnegative matrices. In
the multipartite case, we prove that any Hermitian Matrix Product Density
Operator (MPDO) of bond dimension two is separable, and can be written as a sum
of at most four positive semidefinite matrices per site. This implies that
these states can only contain classical correlations, and very few of them. In
contrast, MPDOs of bond dimension three can contain an unbounded amount of
classical correlations.
