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

## TL;DR

This paper explores various decompositions of mixed states in one dimension, revealing their connections to nonnegative matrix factorizations and providing a comprehensive characterization of these decompositions.

## Contribution

It establishes a correspondence between mixed state decompositions and matrix factorizations, offering new insights into their structure and relationships.

## Key findings

- Decompositions correspond to specific matrix factorizations.
- Symmetric decompositions relate to symmetric matrix factorizations.
- Characterization of mixed state decompositions via matrix theory.

## Abstract

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.