Genuine Multipartite Network Nonlocality in Stabilizer Networks*

Quantum entanglement shared across a multi-node network can be used to generate complicated forms of correlations through local measurements at each node. Some forms of nonlocality exhibited in these configurations are unique to networks and cannot be generated otherwise. We define genuine network k-nonlocality as correlations that can be generated using just (k-1)-partite sources of shared entanglement, but cannot be simulated with classical states. While it is known that genuine network nonlocality (GNN) can emerge in triangle networks, the necessary and sufficient conditions to obtain such correlations remain unspecified. In this work, we show that GNN fails to emerge in multi-qubit triangle networks with no inputs when the networks are limited to stabilizer operations, i.e. state preparations/measurements in the computational basis along with Clifford gates. We also present numerical evidence that GNN does not emerge in triangle networks with local input choices of dichotomic stabilizer measurements and independent EPR states shared between pairs of nodes. Hence, just as non-stabilizer operations are needed to achieve universal quantum computation, we show that non-stabilizer operations are also needed to achieve genuine network nonlocality in multi-qubit triangle networks.