# Separability for mixed states with operator Schmidt rank two

## TL;DR

This paper proves that bipartite mixed states with operator Schmidt rank two are separable and can be expressed as sums of positive semidefinite matrices, using tools from operator theory and matrix decompositions.

## Contribution

It provides a new proof of separability for states with operator Schmidt rank two and extends the result to multipartite states with bounded classical correlations.

## Key findings

- States with operator Schmidt rank two are separable.
- Multipartite states with bond dimension two are separable and have limited classical correlations.
- States with bond dimension three can have unbounded classical correlations.

## Abstract

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.