Covariance matrix-based criteria for network entanglement

TL;DR

This paper develops analytical criteria based on covariance matrices to identify entanglement in quantum networks, enhancing understanding of correlations generated by quantum sources and local operations.

Contribution

It introduces new decompositions and criteria for covariance matrices, enabling practical detection of entanglement in bipartite quantum networks.

Findings

Derived simple proofs for covariance matrix decompositions.

Established necessary criteria for network state preparation.

Applicable to bipartite networks with at most one source per node.

Abstract

Quantum networks offer a realistic and practical scheme for generating multiparticle entanglement and implementing multiparticle quantum communication protocols. However, the correlations that can be generated in networks with quantum sources and local operations are not yet well understood. Covariance matrices, which are powerful tools in entanglement theory, have been also applied to the network scenario. We present simple proofs for the decomposition of such matrices into the sum of positive semidefinite block matrices and, based on that, develop analytical and computable necessary criteria for preparing states in quantum networks. These criteria can be applied to networks in which any two nodes share at most one source, such as all bipartite networks.