Heralded long-distance entanglement schemes for waveguide systems in quantum networks

Establishing quantum entanglement between pairs of distant quantum memories (QMs) is essential for numerous applications in quantum networks. In this paper, we propose some heralded schemes for entanglement generation between multiple pairs of QMs placed in long-distance separated nodes, where the qubits are encoded on QMs coupled to a one-dimensional waveguide. By distributing a pair of spatial-polarization hyperentangled photons to two distant nodes, where each photon scatters with QMs, we can generate entanglement between multiple pairs of distant QMs. In our schemes, the faulty scattering events can be filtered out and discarded via detecting the polarization states of scattered photon components within each module QM. Moreover, the responses from single-photon detectors placed at the end of each node enable us to obtain direct entangled states between QMs in a heralded manner. Eventually, we theoretically extend the proposed schemes by demonstrating specific regularities that arise when entangling more pairs of QMs. With recent advancements in hyperentanglement generation, our schemes become increasingly feasible for practical implementation.