Mixed states in one spatial dimension: decompositions and correspondence with nonnegative matrices

Understand

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.