General theory of sending-or-not-sending twin-field quantum key distribution

The Sending-or-Not-Sending (SNS) protocol of the twin-field quantum key distribution (TF-QKD) has its advantage of unconditional security proof under any coherent attack and fault tolerance to large misalignment error even considered finite-key effect. Here we present a general theory that drastically improves the performance of the SNS protocol. We use the refiened structure of bit-flip error rate and error rejection through parity check with two-way classical communications. This drastically improves the performance of the SNS protocol for the twin-field quantum key distribution. In particular, given misalignment error of $6\%$, our improved SNS protocol exceeds the linear scale of key rate even when the dark count rate is as large as $10^{-6}$, and exceeds a secure distance of 500km even the dark count rate is $10^{-8}$.