- Chinese Optics Letters
- Vol. 20, Issue 9, 092701 (2022)
Abstract
1. Introduction
The first quantum key distribution (QKD) protocol, BB84 protocol[
Reference frame misalignment is regarded as another inevitable problem of the QKD system, such as phase drift in the phase encoding scheme, which plays a severe negative role in disturbing the stable operation of QKD systems. By preparing and measuring the states by one more basis, the reference frame independent (RFI) protocol[
One of the assumptions in guaranteeing the security of MDI-QKD is the perfect preparation of quantum signals, which is not rigorous since perfect preparation devices in realistic scenarios do not exist. In Ref. [25], the impact of imperfect states is investigated in two types of phase encoding MDI-QKD schemes, and the rigorous estimation of secure key rates is given by the quantum coin (QC) method. Additionally, Ref. [26] proposed an improved and rigorous method to consider the basis dependent coding errors in MDI-QKD, where precise source coding can be loosened. Subsequently, the loss-tolerant (LT) protocol[
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2. Protocol
We introduce the RFI-MDI-QKD protocol based on the polarization multiplexing phase encoding scheme[
Figure 1.Schematic diagram of phase encoding polarization multiplexing MDI-QKD protocol. BS, beam splitter; PBS, polarization beam splitter; PMZI, polarization-multiplexing Mach–Zehnder interferometer; PC, polarization controller and compensation. The polarization maintaining fibers separated by PBS are colored red and green to represent the orthogonal polarization H and V, respectively. The case is depicted by pulse H (red) and pulse V (green) when |ϕ0X〉A and |ϕ0Z〉B are prepared.
The role of Charlie is to detect a successful coincidence of quantum states, which are projected into the Bell state , , which corresponds to the case where they send the same or converse bit of the Z basis. cannot be distinguished and used for the key generation because two indistinguishable photons will meet at the beam splitter (BS) simultaneously, and Hong–Ou–Mandel (HOM) interference occurs[
Figure 2.Detector result. The figure illustrates how the double click of Charlie’s site detector (DH0, DH1, DV0, DV1 shown in Fig.
Here, and represents the total number of pulses that are sent from Alice and Bob with the intensity choices . Superscripts and indicate the upper and lower bounds of the corresponding quantity. Further, the corresponding single-photon yield can be ranged by
Meanwhile, parts of gains are classified into error gains . As for the Z basis, , and single-photon error rate , we can see . Similarly, the error rate of the other basis choice can find its lower or upper bound.
Above all, after is sacrificed for error correction, the final key of RFI-MDI-QKD can be bounded by[
Notice that is crucial, in which the RFI protocol takes effect and keeps the key rate vulnerable to the reference frame drift. When considering finite-key effect, the minimum of is reckoned by choosing the corresponding value between its upper and lower bounds:
When source flaws are considered, the phase error rate above is not simply the error rate derived from but given by the yield of fictitious or virtual states . Based on the LT protocol, the sending of Alice’s states to Eve (Charlie) prepared in the Z basis can be equivalently written into the entanglement state, which is
Similarly, the fictitious state of Bob can be also deduced as Alice’s. Subsequently, the yields when Alice and Bob send their fictitious states can be expressed as
Here, focusing on Alice, denotes the probability of emitting Alice’s fictitious state calculated by that includes four elements, which is the Bloch vector of virtual states calculated by , where means the corresponding density matrix of state, and is Pauli matrices. For Bob, and are deduced the same way. In the end, indicates the transmission rate of representing the composite system of transmission channels in the MDI-QKD protocol, which is Alice to Eve and Bob to Eve. Shown as Eq. (11) and elaborated in Section 3, the real state or can be quantified, which further implies , , , and can also be quantified. However, still remains unknown. The solution is to calculate them from the yield of actual states in a form similar to Eq. (9), which is
The corresponding value for real states , , , and can be quantified as well. Here, can be regarded as an unknown 16-element aggregation. Therefore, only four states for the preparation of Alice and Bob, which are , are sufficient to form a system of linear equations to solve . As the mathematics relationship between the yield of actual and fictitious states implies, the yield of fictitious states can also find their bounds by substituting the upper and lower bounds into Eq. (9). By this method, we could deduce fictitious phase error as well as corresponding with the influence of source flaw and finite-key effect in the final key rate simulation.
3. Source Flaw
As the protocol requires, it is necessary to quantify while considering the source flaws and phase drift. In the following, the phase drift is denoted by characterizing the relationship between reference frames of the X and Y basis, which is and . Next, for the source flaw, defines the deficiency in preparation of and is mainly derived from the finite extinction ratio of IM2. In addition, represents the imperfection due to asymmetrical attenuation between two arms of the PMZI. Also, characterizes the imperfection of phase modulation on . The method to quantify the source flaws is to calibrate the derivation between a perfect preparation structure and an imperfect one. However, perfect modulation does not exist. Therefore, in the MDI-QKD protocol, a more rigorous approach is to quantify the source flaw of derivation between Alice and Bob rather than that of each site. It is equal to a situation where the Alice site is perfect, and all imperfections are attributed to the Bob site, which is actually the derivation of source flaw as well as the phase drift between Alice and Bob. In this way, the single-photon states of Alice are represented by
Subsequently, the gains of detectors or are determined by the interference of coherent states arriving at BS1(2). Considering the case where both Alice and Bob prepare 0(1) bit in the Z basis, Alice’s state with intensity arriving at BS1 and BS2 can be represented by and zero. The total loss after the fiber channel distance is estimated by , where and denote the fiber channel attenuation coefficient and detector efficiency, respectively. For Bob, it corresponds to and . As for the case when is prepared by both sites, and represent the states at BS1. Meanwhile, and arrive at BS2. According to Ref. [32], we define () as the coherent states of Alice (Bob) arriving at BS1(2), and, thus, the gains of corresponding detectors can be estimated by
4. Analysis
In the beginning, the transmission performance of the RFI-MDI-QKD protocol by the LT and QC methods is illustrated in Fig. 3. For simplicity, are assumed to be the same value , instead of being considered individually. It can be seen that, the key rate given by QC method[
Figure 3.Comparison of RFI-MDI-QKD with loss-tolerant (LT, solid line) and quantum coin (QC, dashed line) methods with different data sizes ( NMaMb = 1011, 1012, 1013) and the source flaw (δ = 0 or 0.075). The intensities of signal and decoy states are optimized. Other simulation parameters are provided in Table
1.2 | 0.2 dB/km | 15% |
Table 1. Simulation Parameter
Then, the key rates at the fixed distance and are studied in Fig. 4, where the X axis is source flaw and phase drift . To indicate the influence of and explicitly, there is only one variable for each key rate curve, in other words, phase drift is set to zero when the change of source flaw is investigated and vice versa. The discussion can be summarized into the following three points. (1) The decrease of key rates due to in the RFI-MDI-QKD protocol is very slight, which demonstrates its strong robustness against reference frame misalignment. Notice that source flaw is the imperfection of phase encoding and has the same effect of as Eq. (11) implies. Therefore, imperfect phase modulation can be tolerated as well. (2) The enlarging gap of key rates under the same source flaw and phase drift reveals that the LT protocol can provide better immunity at a longer transmission distance. (3) The tolerating limitations of source flaw are not finite. This is because the key rate is restricted by growing with the growth of . The upper bound in our simulation has already covered the value of former experimental research[
Figure 4.Key rate versus source flaw δ and phase drift ω in the RFI-MDI-QKD protocol with LT (solid line) and QC (dashed line) methods at the fixed distance of 2 km (red), 20 km (purple), and 40 km (blue).
The joint impact of both source flaws and phase drift on the LT RFI-MDI-QKD is presented by Fig. 5(a). It is noted that the key rate is symmetric about the phase drift of . Concretely, the phase drift of leads to the severest error. However, the key rate cannot remain stable at near the tolerating limitation when the phase drift goes around . In the end, to demonstrate the benefit of the RFI-QKD protocol more clearly, we also compare RFI-MDI-QKD with the original LT MDI-QKD protocol[
Figure 5.(a) Key rate of loss-tolerant RFI-MDI-QKD under the joint impact of source flaw δ and phase drift ω ∈ [0, π/2] at 2 km. (b) Key rate ratio (R1/R2) of MDI-QKD to RFI-MDI-QKD protocol with the LT method, which is denoted by R1 and R2, respectively.
5. Conclusion
In summary, we demonstrate the advantages of RFI-MDI-QKD and provide rigorous key rate estimation with source flaws under finite-key analysis. The protocol inherits the merits of four states preparation of the initial LT MDI-QKD protocol[
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