Memory-assisted long-distance phase-matching quantum key distribution

Abstract

We propose a scheme that generalizes the loss scaling properties of twin-field or phase-matching quantum key distribution (QKD) related to a channel of transmission ${\ensuremath{\eta}}_{\mathrm{total}}$ from $\sqrt{{\ensuremath{\eta}}_{\mathrm{total}}}$ to $\sqrt[2n]{{\ensuremath{\eta}}_{\mathrm{total}}}$ by employing $n\ensuremath{-}1$ memory stations with spin qubits and $n$ beam-splitter stations including optical detectors. Our scheme's resource states are similar to the coherent-state-based light-matter entangled states of a previous hybrid quantum repeater, but unlike the latter our scheme avoids the necessity of employing $2n\ensuremath{-}1$ memory stations and writing the transmitted optical states into the matter memory qubits. The full scaling advantage of this memory-assisted phase-matching QKD (MA-PM QKD) is obtainable with threshold detectors in a scenario with only channel loss. We mainly focus on the obtainable secret-key rates per channel use for up to $n=4$ including memory dephasing and for $n=2$ (i.e., $\sqrt[4]{{\ensuremath{\eta}}_{\mathrm{total}}}$-MA-PM QKD assisted by a single memory station) for error models including dark counts, memory dephasing and depolarization, and phase mismatch. By combining the twin-field concept of interfering phase-sensitive optical states with that of storing quantum states up to a cutoff memory time, distances well beyond 700 km with rates well above ${\ensuremath{\eta}}_{\mathrm{total}}$ can be reached for realistic, high-quality quantum memories (up to 1-s coherence time) and modest detector efficiencies. Similarly, the standard single-node quantum repeater, scaling as $\sqrt{{\ensuremath{\eta}}_{\mathrm{total}}}$, can be beaten when approaching perfect detectors and exceeding spin coherence times of 5 s; beating ideal twin-field QKD requires 1 s. As for further experimental simplifications, our treatment includes the notion of weak nonlinearities for the light-matter states, a discussion on the possibility of replacing the threshold by homodyne detectors, and a comparison between sequential and parallel entanglement distributions.