Phase-matching quantum key distribution with advantage distillation

Phase-matching quantum key distribution (PM-QKD) provides a promising solution to surpass the fundamental rate–distance bound without quantum repeaters. In this paper, we insert an additional advantage distillation (AD) step after quantum communication to improve the performance of PM-QKD. Simulation results show that, by splitting the raw key into blocks of only a few bits so as to identify highly correlated bit pairs, the AD method can tolerate high system misalignment errors and improve the secret key rate and transmission distance significantly, which is very promising in current PM-QKD systems.


Introduction
Quantum key distribution (QKD) [1,2] can distribute information-theoretic secret keys between two legitimate peers Alice and Bob, even in the presence of an eavesdropper Eve. Owing to the advantage of the theoretic security, lots of QKD demonstrations with high rate and long distance have been reported [3][4][5][6][7]. However, the fundamental rate-distance bound [8,9] limits the performance of these demonstrations, which was widely believed to hold for practical QKD systems without relays.
In this paper, we propose to improve the performance of PM-QKD with the advantage distillation (AD) method. Note that the AD method has already been adopted in various QKD protocols [27][28][29][30][31][32][33]. By splitting the raw key into blocks of only a few bits, we can separate highly correlated bit pair from weakly correlated information. Simulation results show that, PM-QKD with AD can tolerate high system misalignment errors, and improve both the secret key rate and transmission distance significantly.
The rest of this paper is organized as follows. In section 2, we briefly review the security of QKD with AD. In section 3, we introduce PM-QKD with AD. In section 4, we illustrate the numerical results of PM-QKD with AD. We conclude the paper in section 5.

Security of QKD with AD
In an entanglement-based scheme, Alice prepares the quantum state 1 √ 2 |00 + |11 and sends the second qubit to Bob through the quantum channel. Then Alice and Bob take inputs from four dimensional Hilbert spaces H A ⊗ H B to apply the measurements of the Z and X bases, where the Z basis consists of |0 and |1 , and the X basis consists of |+ = 1 √ 2 |0 + |1 and |− = 1 √ 2 |0 − |1 . Since the quantum channel is controlled by Eve, the final state shared between Alice and Bob can be given as where Obviously, the single photon error rates in the Z and X bases are constrained by λ 2 + λ 3 = e z and λ 1 + λ 3 = e x , respectively. The secret key rate shared between Alice and Bob can be given by where E is Eve's ancillary state, and S(ρ) = −tr(ρ log ρ) are entropy functions. Note that Eve is free to choose optimal λ i to eavesdrop as long as λ i is constrained by the error rates in different bases.
To improve the performance of QKD, the AD method was proposed [27], which can increase the correlations of raw key bits between Alice and Bob. Specifically, Alice and Bob split their raw key into blocks . Then, they keep the first bit of their initial block as raw key. Obviously, the successful probability of AD on a certain block of size b is and the quantum state shared between Alice and Bob can be given bỹ Consequently, the secret key rate of QKD enhanced with AD can be modified as

PM-QKD with AD
Based on the original PM-QKD proposed in [11,19], we give the procedure of PM-QKD with AD as follows: (b) Measurement. Alice and Bob transmit their quantum states to a third party Eve. If Eve is cooperative, she interferes the states on a 50:50 beam splitter, directs the two output pulses to two threshold detectors L and R, and records her measurement results. (c) Announcement. Eve announces her measurement results. Then, Alice and Bob announce their random phases and intensities j a , μ a and j b , μ b respectively, and keep those events with μ a = μ b = μ i and j a = j b or j a = j b ± M/2. (d) Sifting. Alice and Bob repeat the above steps sufficient times. When Eve announces a successful detection (either L clicks or R clicks), Alice and Bob keep k a and k b as their raw key bits. Bob flips his key bit k b if Eve announces detector R clicks. If j a = j b ± M/2, Bob flips his key bit k b . (e) Parameter estimation. For all the raw data they have obtained, Alice and Bob can estimate the gains Q μ i and quantum bit error rates E μ i for different intensities. Then, they can estimate the information leakage E X μ . (f) AD. For the events with μ a = μ b = μ, Alice and Bob perform AD on every block of b bits in their raw key to obtain highly correlated raw key bits. (g) Post-processing. Alice and Bob perform key reconciliation and privacy amplification on the raw key data to get the final secret keys.
The secret key rate of PM-QKD with AD is subject to where E X μ and E X μ denote the low and upper bounds of E X μ [19], which can be estimated with the decoy-state method [34][35][36], and f denotes the error correction efficiency. We emphasize that, if Alice and Bob adopt infinite number of decoy states, they can obtain the exact values of E X μ , and the second inequality in equation (8) will be reduced to an equality, that is, E X μ = λ 1 + λ 3 .

Simulation
For a typical implementation PM-QKD [19], we assume that the detection efficiency of single-photon detectors η d is 20%, the dark count rate of single-photon detectors p d is 1 × 10 −8 , the error correction efficiency f is 1.1, the loss coefficient of the standard fiber link α is 0.2 dB km −1 , and the number of phase slices D is 16 which is large enough to ignore the effect of discrete phase randomization [19]. For simplicity, we adopt infinite number of decoy states in our simulation, and the calculation models for Q μ , E μ and E X μ are given by [19] With equations (7)-(9), we simulate the performance of PM-QKD with AD in the asymptotic case under different system misalignment errors e d .
The results of PM-QKD with AD under e d = 1% and 13% are shown in figure 1, and the results of PM-QKD without AD are plotted in comparison. Note that, for fair comparison, the simulation of original PM-QKD here only considers the same and opposite phases, which is different from [19]. In figure 1, PM-AD denotes PM-QKD with AD, and PM denotes the original PM-QKD protocol. It can be seen that PM-QKD with AD is more tolerable to noises, and the corresponding optimal values of b are illustrated in figure 2. Under e d = 1%, the secret key rates of PM-AD and PM are the same when the transmission distance is less than about 500 km, and the corresponding optimal b for PM-AD stays to be 1, which indicates that the AD procedure is actually not performed; while PM-AD begins to show its obvious advantage when the transmission distance is larger than about 500 km. Under e d = 13%, PM-AD exhibits  performance better than that of original PM-QKD especially in the long distance range, which demonstrates the necessity of the AD procedure in PM-QKD under large system misalignment error.
To further investigate the great tolerance of noises in PM-QKD with AD, we plot the results of secret key rate under e d = 20% in figure 3 and the corresponding optimal b in figure 4. It can be seen that, under e d = 20%, PM-AD can just beat the Pirandola-Laurenza-Ottaviani-Banchi (PLOB) bound, which demonstrates the necessity of the AD procedure in PM-QKD again. Moreover, as shown in figures 2 and 4, when the transmission distance is not so long, the optimal b stays to be 1 under e d = 1% or 2 under e d = 13%, 20%; while with the increase of transmission distance which will introduce more noises, the correlation of raw key becomes weaker, hence the optimal b begins to increase. We emphasize that in the PM-QKD protocol with AD, larger b promises longer transmission distance and greater noise tolerance, while it also means lower secret key rate. Hence, to accelerate the computation time, we reasonably set the searching range of b to be the set [1,7] in all our simulations.

Conclusion
In conclusion, we have proposed to insert an additional AD procedure in PM-QKD, and investigated its performance in the asymptotic case. Simulation results demonstrate that PM-QKD with AD can tolerate large system misalignment errors only by splitting the raw key of Alice and Bob into blocks of a few bits.
Particularly, the AD procedure does not change the hardware of PM-QKD, which can be conveniently applied to current systems. We expect our work can provide a valuable reference for researchers to design QKD systems.