Based on the principles of quantum mechanics, quantum key distribution (QKD) provides a way of generating information-theoretic secure keys between two distant parties, Alice and Bob. Since the first BB84 protocol^[1] was proposed, much effort has been made to improve practical performance of QKD systems.^[2–4] Many QKD experiments have been successfully demonstrated either in laboratories or field environments,^[5–13] paving the way towards large-scale commercialization of QKD systems.

In most QKD systems, a shared reference frame is needed for Alice and Bob, which inevitably increases the cost and makes the QKD system more complicated. Fortunately, the reference-frame-independent QKD (RFI-QKD) protocol^[14] was proposed by Laing et al., which can generate secret keys even with slowly drifted reference frames. Up to date, there are plenty of theoretical and experimental works were carried out on RFI-QKD.^[15–21] Moreover, due to the security property of MDI protocol, MDI-QKD has attracted more and more attention recently. Combined with RFI-QKD, there are some experiments on RFI-MDI.^[22,23]

On the other hand, deterministic single-photon sources are still not available, and weak coherent laser pulses are usually used in most QKD systems. It brings a chance for eavesdropper (usually called Eve) to exploit the photon-number-splitting (PNS) attack.^[24–26] To combat the PNS attack, the decoy-state method^[2,27–29] has been adopted in practical QKD systems, where different number of decoy states can be employed. Recently, Rusca et al.^[30] did investigation on one-decoy state BB84 QKD. In their work, it is demonstrated that the one-decoy state method can show advantages for most experimental settings.

In this paper, different from Rusca et al.^[30] work which focused on BB84 protocol, we extend one-decoy state method to RFI-QKD and propose a scheme of one-decoy state RFI-QKD, and further compare its performance with two-decoy state correspondence by taking statistical fluctuation analysis into account. Simulation results demonstrate that our one-decoy state RFI scheme can achieve comparable performance as two-decoy state correspondence in terms of secret key rates and transmission distances.

In the one-decoy state RFI-QKD, Alice adopts two intensities (λ ∈ {μ,v}) and three orthogonal basis (Z_A, X_A, and Y_A) to prepare quantum states, then Bob measures those states with basis Z_B, X_B, and Y_B. Generally speaking, the Z basis can be well aligned for common encoding schemes, and the X and Y basis are allowed to drift slowly with an unknown angle β. That is, the Z, X, and Y basis satisfy Z_B = Z_A, X_B = cosβ X_A + sinβY_A, and Y_B = cosβ Y_A – sinβ X_A. Usually, the Z basis is used to generate the final secret keys, the X and Y bases are used to estimate Eve’s information.

After the quantum communication phase, the practical detection probability and quantum-bit error-rate (QBER) with intensity λ of quantum states that Alice sends in ξ_A basis and Bob measures in ξ_B basis can be given as

Considering the post-processing speed of a QKD system mainly depends on the block size being processed, i.e., the number of pulses detected by Bob in Z basis within certain time window, here in our analysis, we assume the block size with a fixed value (n_ZZ). We can get the corresponding number of pulses that need to be sent (N_tot) by Alice with

Then we can get the number of detection n_ξAξB and the number of errors m_ξAξB corresponding to basis ξ_Aξ_B

Through the following equation, we can calculate the fraction of n_ξAξB corresponding to a certain intensity λ

In the following, let us take the finite-data size effect^[30,32] into account. We can obtain the relation between the observed variables () and the corresponding asymptotic case () by Hoeffding’s inequality as follows:

According to Ref. [30], the upper bound and the lower bound of () are given by

The lower-bound of single-photon events, the lower-bound and the upper-bound of vacuum events in basis _ξAξB can be estimated with the following formulae

The upper bound of single-photon error rate , and the upper bound of single-photon bits errors in the _ξAξB basis can be expressed as follows:

In an RFI-QKD, an intermediate quantity C can be used to monitor Eve’s information, given by

As we know, the formula on calculating the secret keys of BB84 protocol has been addressed in Refs. [30,32]

Through similar method, we can deduce out the formula on calculating the secret key rate for RFI-QKD as

In this section, we first simulate the value of C for both one-decoy and two-decoy state RFI-QKDs under different rotation angles (β), and further do comparison between their key generation rates under the same experimental conditions. Finally, we investigate the effect of the finite-data size on one-decoy RFI-QKD, we study the performance of the one-decoy state RFI-QKD with different block sizes (n_ZZ). It should be noted that, the efficiency of the detector and the internal losses of Bob’s apparatus are included in the value of global attenuation η in our simulations. The experimental parameters used for simulations are all listed in Table 1.

Table 1.

Table 1.

Table 1. List of device parameters for simulations. Y0 is the dark count rate of single-photon detectors; ed is the misalignment error rate of optical systems; f is the error correction efficiency; εsec and εcor correspond to privacy and correctness parameters. . Y0 ed f εsec εcor 10−8 0.01 1.16 10−9 10−15

Table 1. List of device parameters for simulations. Y0 is the dark count rate of single-photon detectors; ed is the misalignment error rate of optical systems; f is the error correction efficiency; εsec and εcor correspond to privacy and correctness parameters. .

In Fig. 1, we investigate the value of C in one-decoy state and two-decoy state RFI-QKDs with different relative rotation angles of reference frames (β ∈ {0°, 10°, 20°, 30°}). The performance of both schemes deteriorates with the increase of β, which is attributed to the smearing of correlations of and , , , , and thus poor estimation of C. As we can see from Fig. 1, when the relative rotation angle is small, the values of C in the one-decoy state RFI-QKD is almost the same as that in the two-decoy state method. While when the rotation angle is getting larger, i.e., > 20°, the value of C in the one-decoy state scheme becomes lower than that in the two-decoy state method, representing worse parameter estimations of the one-decoy state scheme under this circumstance.

Fig. 1.

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	Fig. 1. Values of C for one-decoy state RFI-QKD at different relative rotation angles (β ∈{0°, 10°, 20°, 30°}). The lines from top to bottom denote the results of β = 0°, 10°, 20°, and 30°, respectively. Here we set block size nZZ = 106.

Figure 2 shows comparisons of the key rate between our present one-decoy state RFI-QKD and former two-decoy state RFI-QKDs under different rotation angles of reference frames. For general, we did not consider the dead time of single-photon detectors c_DT as in Ref. [30], which mainly affects the high-speed systems on the key generation rate in the initial global attenuation. Simulation results show that our present results can exhibit better performance than the latter when relative rotation angle β is small (< 20°), benefiting from the less sensitive finite size effect in the one-decoy state method. While as the relative rotation angle β is getting larger (>20°), the key rate of the latter scheme exceeds the former, which is mainly due to the worse parameter estimations of the one-decoy state scheme, as indicated with the C value in Fig. 1. That is, with the relative rotation angle β is getting larger, the correlations of , , , and are smearing, which results in the C value in the one-decoy state RFI-QKD getting worse faster than that in the two-decoy state RFI-QKD, and will affect the key generation rate.

Fig. 2.

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	Fig. 2. Comparison of the key rate between the one-decoy state and the two-decoy state RFI-QKDs at different rotation angles β, where the SKR difference denoted by (R1 – R2)/R2. Lines from top to bottom represent β = 0°, 10°, 20°, and 30°, respectively. Here we set block size nZZ = 106.

In order to investigate the effect of the finite-data size on the one-decoy RFI-QKD, we compare the performance of the one-decoy state RFI-QKD with different sizes of block while setting the relative rotation angle with 20° (n_ZZ ∈ {10⁵, 10⁶, 10⁷, 10⁸}), as shown in Fig. 3. Obviously, the performance of the one-decoy state RFI-QKD will increase when the block size is getting larger. While when the block size is relatively larger, e.g., > 10⁷, the key rate is not sensitive to the finite data-size effect. Therefore, proper block sizes should be chosen in practical applications of the present scheme.

Fig. 3.

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	Fig. 3. Key rates versus global attention for the one-decoy RFI-QKD with different block sizes (nZZ ∈ {105, 106, 107, 108}). Lines from bottom to top denote the results of nZZ = 105, 106, 107, 108, individually. Here we set β = 20°.

In summary, we have proposed a scheme on one-decoy state reference-frame-independent quantum key distribution, and investigate its performance under realistic environments. Numerical simulation results show that, the performance of our present scheme can exceed traditional two-decoy state RFI-QKD when the rotation angle of reference frame is not larger, e.g., < 20°. Moreover, it does not need to prepare the vacuum state in the former, and thus reduce the experimental complexity of QKD systems. Therefore, our present work could provide valuable references for practical implementation of the RFI-QKD.