Separability Criterion for multipartite quantum states based on the Bloch representation of density matrices

TL;DR

This paper introduces a new necessary separability criterion for multipartite quantum states based on the Bloch representation and multilinear algebra, capable of detecting entanglement and bound entangled states across various partitions.

Contribution

It presents a novel separability criterion applicable to arbitrary N-partite states, utilizing tensor matrization, and provides conditions for separability and entanglement detection, including for N-qubit PPT states.

Findings

Can detect entanglement in multipartite states

Able to identify bound entangled states

Provides a necessary and sufficient condition for certain N-qubit states

Abstract

We give a new separability criterion, a necessary condition for separability of $N$-partite quantum states. The criterion is based on the Bloch representation of a $N$-partite quantum state and makes use of multilinear algebra, in particular, the matrization of tensors. Our criterion applies to {\it arbitrary} $N$-partite quantum states in $\mathcal{H}=\mathcal{H}^{d_1}\otimes \mathcal{H}^{d_2} \otimes ... \otimes \mathcal{H}^{d_N}.$ The criterion can test whether a $N$-partite state is entangled and can be applied to different partitions of the $N$-partite system. We provide examples that show the ability of this criterion to detect entanglement. We show that this criterion can detect bound entangled states. We prove a sufficiency condition for separability of a 3-partite state, straightforwardly generalizable to the case $N > 3,$ under certain condition. We also give a necessary and sufficient condition for separability of a class of $N$-qubit states which includes $N$-qubit PPT states.