Practical reference-frame-independent quantum key distribution systems against the worst relative rotation of reference frames

Reference-frame-independent quantum key distribution (RFI-QKD) can generate secret keys with the slow drift of reference frames. However, the performance of practical RFI-QKD systems deteriorates with the increasing drift of reference frames. In this paper, we mathematically demonstrate the worst relative rotation of reference frames for practical RFI-QKD systems, and investigate the corresponding performance with optimized system parameters. Simulation results show that practical RFI-QKD systems can achieve quite good performance against the worst relative rotation of reference frames, which exhibit the feasibility of practical QKD systems with free drifting reference frames. Furthermore, we propose a universal estimation method of the secret key rate in practical RFI-QKD systems, which conforms to the nature of RFI-QKD more well than the usual estimation method.


Introduction
Based on the theory of quantum physics, quantum key distribution (QKD) [1] provides a way of generating information-theoretic secure keys between two distant parties Alice and Bob, even in the presence of a malicious eavesdropper Eve. Since the first Bennett-Brassard-1984 (BB84) protocol was proposed in 1984, a series of works have been made to improve the practical performance of QKD systems [2][3][4][5][6][7][8][9][10][11][12][13][14]. Generally, in most of these systems, the complicated and real-time operation of calibrating reference frames is indispensable to assure the regular running of practical QKD systems [4,[11][12][13][14]. However, the calibration of reference frames may complicate QKD systems and reduce key generation rates.
Fortunately, reference-frame-independent QKD (RFI-QKD) [15] can bypass the calibration of reference frames and generate secret keys with the slow drift of reference frames. In RFI-QKD, Alice (Bob) adopts three orthogonal bases Z, X, and Y to encode (decode) quantum states, and her (his) local bases are denoted as Z A(B) , X A(B) , and Y A(B) , where Z, X, and Y are three Pauli matrices. The Z basis can be well aligned for common encoding schemes, and the X and Y bases may drift slowly with unknown angle β. In other words, the Z, X, and Y bases satisfy A . Alice and Bob can distill secret keys from data in Z basis, and estimate Eve's information from data in X and Y bases.
To date, there have been a lot of theoretical and experimental reports on RFI-QKD [16][17][18][19][20][21][22][23][24][25][26] due to its immunity against the slow drift of reference frames. However, the performance of practical RFI-QKD systems worsens with the increasing drift of reference frames, and their performance will become the worst when the relative rotation of reference frames reaches its maximum. Hereafter, the maximum relative rotation of reference frames is termed the worst relative rotation of reference frames. Hence, it is vital to investigate the worst relative rotation of reference frames of practical RFI-QKD systems and the corresponding performance.
In this paper, we mathematically demonstrate the worst relative rotation of reference frames for practical RFI-QKD systems, and investigate the corresponding performance with full optimized parameters. Simulation results show that practical RFI-QKD systems can achieve quite good performance even against the worst relative Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. rotation of reference frames, which obviously exhibit the possibility of practical QKD systems with free drifting reference frames. Moreover, we propose a universal estimation method of the secret key rate in practical RFI-QKD systems, which does not need a good estimation of the relative rotation β.

The worst relative rotation of reference frames for practical RFI-QKD systems
For simplicity, we demonstrate the worst relative rotation of reference frames for RFI-QKD with the weak coherent source without statistical fluctuations. In order to make the demonstration more practical, we introduce the three-intensity decoy-state method [27][28][29] into RFI-QKD, where Alice randomly modulates her quantum states into signal states of intensity μ, decoy states of intensity ν, and vacuum states of intensity ω. For signal (decoy) states of intensity μ(ν), Alice randomly chooses bases Z A , X A , or Y A , and for vacuum states of intensity ω, she does not choose any bases. Bob randomly chooses bases Z B , X B , or Y B to measure the quantum states he received. After the quantum communication phase, Alice and Bob can obtain a series of gains and quantum bit error rates (QBERs).
Here, we give the related system parameters and models used in this demonstration first. d (η d ) denotes the dark count rate (detection efficiency) of single-photon detectors, α (L) represents the loss coefficient (length) of the standard fiber link, h h = -a 10 d L 10 denotes the overall transmission efficiency between Alice and Bob, e d characterizes the intrinsic error rate of the optical system, and f measures the inefficiency of the key reconciliation process. In this paper, we use the typical parameters, which are d=3×10 −6 , η d =14.5% , α=0.2 dB/km, e d =1.5% , and f=1.16, for simulation. Moreover, two threshold single-photon detectors are adopted at Bob's side, and the probability of the valid detection events conditional on j-photon states is where the valid detection events are those only one detector clicks. As for the double-click events in real-life RFI-QKD systems, Bob assigns a random bit value to one detector. The photon-number distribution of the phaserandomized weak coherent source is = where λ is the intensity of the weak coherent source, and i is the number of photons.
Obviously, the gain of vacuum states, denoted as Q ω , is = - , 2 and the QBER of vacuum states, denoted as E ω , is The gain of signal or decoy states of intensity λ prepared in basis ζ A and measured in basis ζ B , denoted as z z l Q A B , can be given as,  (4), Alice and Bob can obtain explicit expressions of different gains. We take l Q X X A B for example to show how to get these explicit expressions. First, l P X X Similarly, we can easily get With the same procedure, we can obtain . With the above gains and QBERs, they can estimate the lower bound of the single-photon yield [29] and the upper bound of the single-photon error rate [29] in ζ A ζ B n = - where H is the binary entropy function given by 1 log 1 2 2 , and I E denotes Eve's information. Particularly, Eve's information is = - As shown in equation (17), the relative rotation of reference frames β only affects Eve's information I E . More specifically, β only affects C. Hence, to investigate the performance of practical RFI-QKD systems against the worst relative rotation of reference frames β, we only need to investigate the performance of C against the worst β. In our demonstration, we consider RFI-QKD with the slow drift of reference frames, where β can be simply deemed as constant in a short period of time. Interestingly, [22,24] assume β suffers from certain variation.
To investigate the performance of C against β, we temporarily set μ=0.6, ν=0.1, and L=100, which obviously will not change the behavior of C against β. First, with equations (16), (21), we can demonstrate C is a periodic function of β by calculations, and the minimal positive period of C is p 2 . In other words, where n=±1,±2, L . Moreover, in the domain 4 4 , which means the symmetry axis of C is p 4 . Figure 1 shows the performance of C against the relative rotations of reference frames β in two periods. It can be easily seen that the minimal positive period of C is p 2 , and the performance of C reaches its worst at the symmetry axis, where β is p 4 or p Then, in the domain p ( ) 0, 2 , we calculate the derivative of C, denoted as ¢ C , and find ¢ = 2 , which illustrates that C reaches the minimum value at b = p 4 . Figure 2 shows the performance of C′ against the relative rotations of reference frames β in p ( ) 0, 2 . Hence, considering the periodicity of equations (17) or (21), we demonstrate the worst relative rotation of reference frames for RFI-QKD is b = p m 4 , where m is odd. The physics behind this demonstration can be explained by means of Bloch sphere: 1) β=0 means that Alice's X A and Y A bases align perfectly with Bob's X B and Y B bases, which is the ideal case; 2) b = p 2 means that Alice's X A is Bob's Y B and Alice's Y A is Bob's X B , which is in essence the ideal case due to the high symmetry of C; and 3) b = p 4 means the worst-aligned reference frames between Alice and Bob, which is the worst relative rotation of reference frames for RFI-QKD. Considering the periodicity of equation (17), we can deduce that the performance of RFI-QKD is worst at b = p m 4 (m is odd).

Simulation
With the above demonstration, we consider the reasonable data set N=10 11 for statistical fluctuations [31] in practical RFI-QKD systems, and investigate its performance in p ⎡ ⎣ ⎤ ⎦ 0, 2 by the full parameter optimization method [32]. Notice that the domain p ⎡ ⎣ ⎤ ⎦ 0, 2 is ample to study the performance of practical RFI-QKD systems due to the periodicity of equations (17) or (21). For comparison purpose, we investigate the performance of practical RFI-QKD with b = p p p p 0, , , 2 . Simulation results are shown in figure 3. In figure 3, the solid lines from right to left denote the secret key rate of RFI-QKD with b = p p 0, , 8 4 , respectively, and the dotted lines  be noted that, to clearly show the performance of RFI-QKD against the worst relative rotation of reference frames, results in figure 3 are simulated with the usual method.

Conclusions
In conclusion, we have demonstrated the worst relative rotation of reference frames for practical RFI-QKD, and investigated its performance with optimized parameters. Simulation results show that RFI-QKD can achieve quite good performance against the worst relative rotation of reference frames, which clearly demonstrates the possibility of practical QKD systems with free drifting reference frames. We believe practical QKD systems with free drifting reference frames will easily find a place in the satellite-to-ground QKD scenario [11,12] or other scenarios where it is difficult to calibrate the reference frames. Furthermore, we propose a universal estimation method of the secret key rate in practical RFI-QKD systems, which conforms to the nature of RFI-QKD more well than the usual estimation method.