  We study six natural decompositions of mixed states in one spatial dimension:
the Matrix Product Density Operator (MPDO) form, the local purification form,
the separable decomposition (for separable states), and their three
translational invariant (t.i.) analogues. For bipartite states diagonal in the
computational basis, we show that these decompositions correspond to
well-studied factorisations of an associated nonnegative matrix. Specifically,
the first three decompositions correspond to the minimal factorisation, the
nonnegative factorisation, and the positive semidefinite factorisation. We also
show that a symmetric version of these decompositions corresponds to the
symmetric factorisation, the completely positive factorisation, and the
completely positive semidefinite transposed factorisation, respectively. We
leverage this correspondence to characterise the six decompositions of mixed
states.
