Entangling Quantum Memories via Heralded Photonic Bell Measurement

Photonically-heralded Bell swaps have been proposed to generate entanglement between a pair of matter qubits, including superconducting, trapped-ion and color center qubits. The quality of the heralded entangled state depends on whether the photonic qubit is encoded in either the single rail (logical states are the presence or absence of a photon in a mode) or the dual rail (logical states are the presence of a single photon in one of two orthogonal modes) format. We evaluate the density operator of the heralded two-qubit entangled state of the matter qubits, as a function of the aforesaid encoding format, the visibility of the interferometer used for the photonic swap, any residual phase error among the two paths, efficiency and dark noise in the single-photon detectors. We show that the entanglement rate of the dual rail scheme scales as O(η), and that of the single-rail scheme scales as O(√η). But, at low loss (√η< 8 dB), the dual rail achieves a higher rate. In addition to Fidelity of the heralded entangled state, we also quantify a lower bound to the distillable entanglement: the number of ideal Bell states extractable per copy of the noisy entangled state, assuming both parties have fault-tolerant quantum processing, and classical communications.