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Twin field quantum key distribution promises high key rates at long distance to beat the rate distance limit. Here, applying the sending or not sending TF QKD protocol, we experimentally demonstrate a secure key distribution breaking the absolute key rate limit of repeaterless QKD over 509 km, 408 km ultra-low loss optical fibre and 350 km standard optical fibre. Two independent lasers are used as the source with remote frequency locking technique over 500 km fiber distance; Practical optical fibers are used as the optical path with appropriate noise filtering; And finite key effects are considered in the key rate analysis. The secure key rates obtained at different distances are more than 5 times higher than the conditional limit of repeaterless QKD, a bound value assuming the same detection loss in the comparison. The achieved secure key rate is also higher than that a traditional QKD protocol running with a perfect repeaterless QKD device and even if an infinite number of sent pulses. Our result shows that the protocol and technologies applied in this experiment enable TF QKD to achieve high secure key rate at long distribution distance, and hence practically useful for field implementation of intercity QKD.Introduction.-Channel loss seems to be the most severe limitation on the practical application of long distance quantum key distribution (QKD) [1-3], given that quantum signals cannot be amplified. Much efforts have been made towards the goal of a longerdistance for QKD [4][5][6]. Theoretically, the decoy-state method [7][8][9] can improve the key rate of coherent-state based QKD from scaling quadratically to a linear with the channel transmittance, as what behaves of a perfect single-photon source. This method can beat the photonnumber-splitting attack to the imperfect single-photon source and the coherent state is used as if only those single-photon pulses were used for key distillation, and hence it can reach the key rate to a level comparable with that of a perfect single-photon source.

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The Sending-or-Not-Sending 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. So far this is the only coherent-state based TF-QKD protocol that has considered finite-key effect, the statistical fluctuations. Here we consider the complete finite-key effects for the protocol and we show by numerical simulation that the protocol with typical finite number of pulses in practice can produce unconditional secure final key under general attack, including all coherent attacks. It can exceed the secure distance of 500 km in typical finite number of pulses in practice even with a large misalignment error. the loophole. A series of experiments [70][71][72][73] have been done to demonstrate those protocols. In particular, an efficient protocol for TF-QKD through sending-or-notsending (SNS protocol) has been given in Ref. [63]. The SNS protocol has been experimentally demonstrated in proof-of-principle in Ref. [70], and realized in real optical fiber with the effects of statistical fluctuation being taken [71]. The unconditional security of SNS protocol in the asymptotic case has been proved [63] and SNS protocol relaxes the requirement for single photon interference accuracy. The key rate of SNS is still considerable even if the misalignment error is as large as 35%. Among all those variants of TF QKD with coherent states, the SNS QKD protocol is the only one that takes the effect of statistical fluctuation and finite decoy states into consideration [68]. Here we show an analysis of the complete effect of finite-key size of SNS QKD protocol.The main tool we use to analyse the effect of finitekey size is the universally composable framework [74]. An complete QKD protocol usually includes the preparation and distribution of quantum states, measurement of received quantum states, parameter estimation, error correction and private amplification. After the error correction step, Alice gets a bit string S, and Bob get an estimate string S ′ of S. If the error rate is too large, the results of error correction is an empty string and the protocol aborts. A protocol is called ε cor -correct if the probability that S and S ′ aren't the same, Pr(S = S ′ ) ≤ ε cor .Besides, the quantum state of Alice may be attacked by Eve in the distribution and measurement steps and some information would be leaked to Eve. To ensure the security of final secret keys, Alice and Bob apply a privacy amplification scheme based on two-universal hashing [75] to extract two shorter strings of length l from S and S ′ . We denote the density operator of the