Entangling Quantum Memories via Heralded Photonic Bell Measurement

A common way to entangle a pair of quantum memories is via a photonic entanglement swap. Each of two memories, connected by an optical channel, emits a photonic qubit entangled with itself, and the photonic qubits undergo an entanglement swap on a beamsplitter in the middle of the channel. We compare two choices of encoding of the photonic qubit: the single rail and dual rail. In the regime of low channel loss, i.e., when the loss of the half-channel connecting one memory site to the swap site is less than $\approx 6$ dB, the dual-rail scheme is seen to outperform the single rail scheme. The high-loss rate asymptote for the dual rail scheme is worse: it scales as $O(\eta)$ ebits per transmitted photonic mode, as opposed to $O(\sqrt{\eta})$ for the single-rail scheme, where $\sqrt{\eta}$ is the transmissivity of the half channel. Considering the following non-idealities: imperfect mode matching at the swap, carrier-phase mismatch across the interfered photonic qubits from the two sides, and detector excess noise, we evaluate the explicit density operator of the heralded two-qubit entangled state. We calculate a lower bound on its distillable entanglement per copy, and its Fidelity (with the ideal Bell state). Imperfect swap-visibility results in a constant-factor decrease in the rate, while excess noise results in a sharp dropoff of distillable entanglement beyond a certain total channel loss threshold to zero. Despite the single-rail scheme's better rate-loss scaling, it is more severely affected by excess noise, and is adversely affected by stochastic carrier-phase mismatch. We study entanglement distillation on the heralded noisy entangled states. Our evaluation of the density operator of the entangled state will hopefully pave the way for more realistic performance evaluations of larger quantum networks and the development of advanced entanglement distillation schemes.