# On decomposable correlation matrices

## TL;DR

This paper introduces the concept of r-decomposable correlation matrices, characterizes their properties, and explores their implications in quantum information theory, including entanglement detection.

## Contribution

It defines r-decomposable correlation matrices, proves key properties, and constructs examples that are not r-decomposable, advancing understanding in quantum correlation structures.

## Key findings

- Every (r+1) x (r+1) correlation matrix is r-decomposable for r ≥ 2
- Constructed (2r+1) x (2r+1) correlation matrices that are not r-decomposable
- Partial progress on whether all 4x4 correlation matrices are 2-decomposable

## Abstract

Correlation matrices (positive semidefinite matrices with ones on the diagonal) are of fundamental interest in quantum information theory. In this work we introduce and study the set of $r$-decomposable correlation matrices: those that can be written as the Schur product of correlation matrices of rank at most $r$. We find that for all $r \geq 2$, every $(r+1) \times (r+1)$ correlation matrix is $r$-decomposable, and we construct ${(2r+1) \times (2r+1)}$ correlation matrices that are not $r$-decomposable. One question this leaves open is whether every $4 \times 4$ correlation matrix is $2$-decomposable, which we make partial progress toward resolving. We apply our results to an entanglement detection scenario.