Exact rate analysis for quantum repeaters with imperfect memories and entanglement swapping as soon as possible

We present an exact rate analysis for a secret key that can be shared among two parties employing linear quantum repeater chain. One of our main motivations is to address the question whether simply placing memories along communication channel beneficial in realistic setting. The underlying model assumes deterministic entanglement swapping single-spin and it excludes probabilistic distillation, thus two-way classical communication, on higher nesting levels. Within this framework, we identify essential properties any optimal scheme: distribution parallel, as soon parallel storage little possible. While these features are obvious or trivial simplest with one middle station, more stations they cannot always combined. propose scheme including loss memory dephasing, proving its optimality case conjecturing general case. In even setting, consider additional tools parameters such cutoffs, multiplexing, initial state gate fidelities, finite link coupling efficiencies order potential regimes memory-assisted beyond station exceed rates smallest repeaters well those obtainable all-optical schemes unassisted by stationary qubits communication. Our analytical treatment enables us determine simultaneous trade-offs between various parameters, their scaling, influence performance ordering different types protocols, comparing two-photon interference after dual-rail qubit transmission one-photon single-rail or, similarly, optical coherent states. find experimental parameter values highly demanding but not impossible (up 10 s coherence time, about 80% coupling, infidelities regime 1%--2%), bit per second at total budget 157.6 dB, i.e. distance 800 km fiber attenuation length 22 placed every 100 km---a clear improvement over twin-field much pronouncedly, ideal point-to-point GHz clock rates.