The BB84 quantum key distribution (QKD) combined with decoy-state method is currently the most practical protocol, which has been proved secure against general attacks in the finite-key regime. Thereinto, statistical fluctuation analysis methods are very important in dealing with finite-key effects, which directly affect secret key rate, secure transmission distance and most importantly, the security. There are two tasks of statistical fluctuation in decoy-state BB84 QKD. One is the deviation between expected value and observed value for a given expected value or observed value. The other is the deviation between phase error rate of computational basis and bit error rate of dual basis. Here, we provide the rigorous and optimal analytic formula to solve the above tasks, resulting to higher secret key rate and longer secure transmission distance. Our results can be widely applied to deal with statistical fluctuation in quantum cryptography protocols. So far, there have existed many kinds of protocols describing how quantum key distribution (QKD) 1,2 works, such as the Bennett-Brassard-1984 (BB84) 3 , Bennett-Brassard-Mermin-1992 4 , Bennett-1992 5 , six-state 6 , continuous variable 7,8 and measurement-device-independent 9-11 protocols. Although different protocols contain different processes, they all serve the same purpose to guarantee that two parties, named Alice and Bob, can share a string of key data through a channel fully controlled by an eavesdropper, named Eve 3. Unlike some computational assumptions, these protocols are all proven to be secure with fundamental physical laws in the recent years 12-18 , which shows the great advantage in information transmitting that QKD holds. BB84 stands out as the most important protocol due to its best overall performance. However, implementations of the BB84 protocol differ from the original theoretical proposal. For an ideal single-photon source is not available yet, in actuality, a weak pulsed laser source is in place of it. Nevertheless, there is a critical flaw in the weak pulsed laser source that an non-negligible part of laser pulses contains more than one photon, which will be exploited by Eve through the photon-number-splitting (PNS) attack 19. To address this drawback with high channel loss, the decoy-state method is introduced 20-22. The source will generate the phase-randomized coherent state in decoy-state method, which can be regarded as the mixed photon number state. The essence of the decoy state idea can be summarized as that the yield (bit error rate) of n-photon in signal state is equal to that in decoy state. However, this equal-yield condition can only be established under the asymptotic-key regime. The expected value of yield (bit error rate) of n-photon in signal state and decoy state are identical while the corresponding observed value cannot be assumed to be the same in the finite-key regime. By exploiting the decoy-state method, one can establish the linear system of equations about the expected values to obtain expected value of yield (bit er...
