We study six natural decompositions of mixed states in one spatial dimension: the matrix product density operator form, the local purification form, the separable decomposition (for separable states), and their three translational invariant analogs. For bipartite states diagonal in the computational basis, we show that these decompositions correspond to well-studied factorizations of an associated nonnegative matrix. Specifically, the first three decompositions correspond to the minimal factorization, the nonnegative factorization, and the positive semidefinite factorization. We also show that a symmetric version of these decompositions corresponds to the symmetric factorization, the completely positive factorization, and the completely positive semidefinite transposed factorization. We leverage this correspondence to characterize the six decompositions of mixed states.
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