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

We present improved method of sending-or-not-sending twin-field quantum key distribution (SNS TF-QKD) based on its structure and the application of error rejection. %And we present iteration formula for bit-flip error rate of the survived bits after error rejection. Taking the finite key effect into consideration with only $10^{11}$ total pulses, we show that the method here exceeds the absolute limit of repeater-less key rate with whatever detection efficiency. We also make comparative study of different protocols numerically. It shows that the method here presents advantageous results at long distance regime and large noise regime, asymptotically or non-asymptotically. Applying the de Finetti theorem, we present the iteration formula of phase-flip error rate for survived bits from odd-parity events only. With this, the performance of SNS protocol can be further improved a lot.