Twin-field quantum key distribution with large misalignment error

Based on the novel idea of twin-field quantum key distribution [TF-QKD; Lucamarini et al., Nature (London) 557, 400 (2018)], we present a protocol named the ``sending or not sending TF-QKD'' protocol, which can tolerate large misalignment error. A revolutionary theoretical breakthrough in quantum communication, TF-QKD changes the channel-loss dependence of the key rate from linear to square root of channel transmittance. However, it demands the challenging technology of long-distance single-photon interference, and also, as stated in the original paper, the security proof was not finalized there due to the possible effects of the later announced phase information. Here we show by a concrete eavesdropping scheme that the later phase announcement does have important effects and the traditional formulas of the decoy-state method do not apply to the original protocol. We then present our ``sending or not sending'' protocol. Our protocol does not take postselection for the bits in $Z$-basis (signal pulses), and hence the traditional decoy-state method directly applies and automatically resolves the issue of security proof. Most importantly, our protocol presents a negligibly small error rate in $Z$-basis because it does not request any single-photon interference in this basis. Thus our protocol greatly improves the tolerable threshold of misalignment error in single-photon interference from the original a few percent to more than $45%$. As shown numerically, our protocol exceeds a secure distance of 700, 600, 500, or 300 km even though the single-photon interference misalignment error rate is as large as $15%, 25%, 35%$, or $45%$.