Symmetries in quantum networks lead to no-go theorems for entanglement distribution and to verification techniques

The analysis of correlations in quantum networks is a difficult problem in the general case, and have so far been limited to small examples. Here, the authors show how to use symmetries and the inflation technique to derive general network entanglement criteria and certification methods. Quantum networks are promising tools for the implementation of long-range quantum communication. The characterization of quantum correlations in networks and their usefulness for information processing is therefore central for the progress of the field, but so far only results for small basic network structures or pure quantum states are known. Here we show that symmetries provide a versatile tool for the analysis of correlations in quantum networks. We provide an analytical approach to characterize correlations in large network structures with arbitrary topologies. As examples, we show that entangled quantum states with a bosonic or fermionic symmetry can not be generated in networks; moreover, cluster and graph states are not accessible. Our methods can be used to design certification methods for the functionality of specific links in a network and have implications for the design of future network structures.