Composable security for practical quantum key distribution with two way classical communication

Abstract

We present a theory for the composable security of sending-or-not-sending (SNS) protocol of twin field quantum key distribution (TF-QKD). We present methods to strictly calculate the finite-key effects in QKD with error rejection through two-way classical communication (TWCC) for SNS TF-QKD protocol. Unlike the normal QKD without TWCC, here the probability of tagging or untagging for each two-bit random group is not independent. We rigorously solve this problem by imagining a virtual set of bits where every bit is independent and identical. With explicit formulas, we show that simply applying Chernoff bound in the calculation gives correct key rate, but the failure probability changes a little bit. We calculate the key rate with strict bounds and security, and obtain key rates by far breaking the PLOB (Pirandola, Laurenza, Ottaviani, and Banchi) bound with composable security.