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

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.