Illustration of phase post-compensation. Without loss of generality, here we consider the case κa = κ b. Denote the phase references of Alice and Bob as φa0 and φ b0 , and hence the reference deviation is φ0 = φ b0 − φa0 mod 2π. Bob can figure out the proper phase compensation offset j d by minimizing the QBER from random sampling as follows. Bob sets up a j d , sifts the bits by |j b + j d − ja| = 0, and evaluates the sample QBER. He tries all possible j d ∈ {0, 1, · · · , M − 1} and figures out the proper j d to minimize the sample QBER. And then he announces the sifted locations of unsampled bits to Alice. As shown in figure, we set M = 12, and the reference deviation φ0 = 70 • , hence Bob can set j d = 2 to compensate the effect of φ0. 

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... address these practical issues, we employ a phase post-compensation method [18], where Alice and Bob first divide the phase interval [0, 2π) into M slices {∆ j } for 0 ≤ j ≤ M −1, where ∆ j = [2πj/M, 2π(j+1)/M ). In- stead of comparing the exact phases, Alice and Bob only compare the slice indexes. This makes the phase sifting step practical, but introduces an intrinsic misalignment error. Besides, Alice and Bob do not perform the phase sifting immediately in each round, and instead, they do it in data postprocessing. In the parameter estimation step, they perform the following procedures, as shown in Fig. 2. 1. For each bit, Alice announces the phase slice index j a and randomly samples a certain amount of key bits and announces them for QBER ...

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... sifting, Bob can estimate the offset j 0 for each pulse. The estimation accuracy would not affect the se- curity. In fact, in the security proofs, we assume Eve knows the phase references ahead. Suppose Bob com- pensates the offset by j d , and he has the freedom to choose j d from {0, 1, · · · , M − 1}. In a practical scenario, normally the phase references (and hence the offset j 0 ) change slowly with time. Then, Bob can figure out the proper phase compensation offset j d by minimizing the QBER from random sampling, as shown in Fig. ...

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... we modify the PM-QKD protocol to remove the requirement of phase locking between Alice and Bob, and also relax the post-selection condition of |φ a − φ b | = 0 or π. The method is shown in Fig. ...

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... and Bob only compare the slice indexes. This makes the phase-sifting step practical, but introduces an intrinsic misalignment error. Also, Alice and Bob do not perform the phase- sifting immediately in each round, and instead, they do it in data postprocessing. In the parameter estimation step, they perform the following procedures, as shown in Fig. 2. 1. For each bit, Alice announces the phase slice index j a and randomly samples a certain amount of key bits and announces them for QBER ...

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... we modify the PM-QKD protocol to remove the requirement of phase locking between Alice and Bob, and also relax the postselection condition of |φ a − φ b | = 0 or π. The method is shown in Fig. 2. ...

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... Suppose Bob compen- sates the offset by j d , and he has the freedom to choose j d from {0, 1, · · · , M − 1}. In a practical scenario, normally the phase references (and, hence, the offset j 0 ) change slowly with time. Then, Bob can figure out the proper phase compensation offset j d by minimizing the QBER from random sampling, as shown in Fig. ...