  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.
