Probability distribution of three-parameter symmetric joint measurement on a triangular network

Abstract Quantum networks relying on entanglement swapping offer practical resources for quantum communication, generally assisted by Bell state measurements (BSM). In contrast to conventional BSM, the novel symmetric joint measurements not only subsume standard BSM but also feature parameterization advantages. We investigate the probability distribution of three-parameter symmetric joint measurements on a triangular network. A primary result is that the output probabilities generally depend on the quantum state distributed by the sources to the three parties. We consider four Bell states serving as initial states, exploring the permutation symmetry of the measurement probabilities while applying them to nonlocality tests in a triangular network. The results demonstrate that the three-parameter symmetric joint measurement is highly effective for network nonlocality testing. Specifically, when using three singlet states as the initial states, the network nonlocality is robustly exhibited regardless of the selection of measurement parameters z , ϕ , and θ .