  To quantify quantum advantage in communication networks, we apply an
operational framework for witnessing quantum nonclassicality. Following
previous approaches in the field, this framework first computes linear
constraints on the input/output probabilities that arise in classical networks
when the amount of communication is bounded. We then apply variational quantum
algorithms to optimize these probabilities when quantum communication resources
are introduced. Any violation of the classical constraints indicates that extra
classical communication is needed to simulate the comparable quantum network,
thereby demonstrating an explicit quantum advantage. We demonstrate
nonclassicality in many basic networks such as entanglement-assisted
point-to-point and multi-point channels. In all examples, we find that
equipping classical or quantum channels with entanglement leads to
nonclassicality, whereas networks having multiple senders do not require
entanglement to achieve nonclassicality. Finally, we discuss how our approaches
could be implemented on quantum networking hardware and used to automatically
establish certain protocols.
