Quantum key distribution allows remote parties to generate information-theoretic secure keys. The bottleneck throttling its real-life applications lies in the limited communication distance and key generation speed, due to the fact that the information carrier can be easily lost in the channel. For all the current implementations, the key rate is bounded by the channel transmission probability η. Rather surprisingly, by matching the phases of two coherent states and encoding the key information into the common phase, this linear key-rate constraint can be overcome-the secure key rate scales with the square root of the transmission probability, O( √ η), as proposed in twin-field quantum key distribution [Nature (London) 557, 400 (2018)]. To achieve this, we develop an optical-mode-based security proof that is different from the conventional qubit-based security proofs. Furthermore, the proposed scheme is measurement device independent, i.e., it is immune to all possible detection attacks. The simulation result shows that the key rate can even exceed the transmission probability η between two communication parties. In addition, we apply phase postcompensation to devise a practical version of the scheme without phase locking, which makes the proposed scheme feasible with the current technology. This means that quantum key distribution can enjoy both sides of the world-practicality and security.

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We found an error in the simulation code in our recent work. With the coding error fixed, the simulation result (Fig. 3 in the original paper) is shown below. The longest distance under the same parameters in Table I is 418 km. As is shown in Fig. 1, the key rate line of PM-QKD can still overcome the secret key capacity bound when l > 250 km, and the rate-loss dependence of PM-QKD, R ∼ Oð ffiffi ffi η p Þ, is not affected. The simulation error was brought to our attention when we ran the simulation in a follow-up work, and it was independently pointed out by Ivan Djordjevic. In the Supplemental Material [1], we present the MATLAB code of the PM-QKD key rate for reference. [1] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevX.9.029901 for the MATLAB code of the PM-QKD key-rate evaluation. FIG. 1. Corrected simulation result of Fig. 3 in the original article. The purple dashed line is the original incorrect plot of the PM-QKD key rate result, which is caused by an erroneous extra term ð2πÞ=M when calculating the QBER using Eq. (B22); see lines 27 and 28 in the MATLAB code file in the Supplemental Material [1].

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