Detecting genuine multipartite correlations in terms of the rank of coefficient matrix

TL;DR

This paper introduces a method to identify genuine multipartite quantum correlations by analyzing the rank of coefficient matrices, providing a necessary and sufficient condition for such correlations.

Contribution

It presents a novel criterion based on matrix rank to detect genuine quantum correlations in arbitrary states, and offers a way to decompose correlated states into uncorrelated product states.

Findings

Genuine quantum correlations correspond to coefficient matrices with rank greater than one.

The method provides a necessary and sufficient condition for the presence of genuine correlations.

Decomposition of high-rank correlated states into uncorrelated product states is demonstrated.

Abstract

We propose a method to detect genuine quantum correlation for arbitrary quantum state in terms of the rank of coefficient matrices associated with the pure state. We then derive a necessary and sufficient condition for a quantum state to possess genuine correlation, namely that all corresponding coefficient matrices have rank larger than one. We demonstrate an approach to decompose the genuine quantum correlated state with high rank coefficient matrix into the form of product states with no genuine quantum correlation for pure state.