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Abstract
Mode-pairing quantum key distribution (MP-QKD) holds great promise for the practical implementation of QKD in the near future. It combines the security advantages of measurement device independence while still being capable of breaking the Pirandola–Laurenza–Ottaviani–Banchi bound without the need for highly demanding phase-locking and phase-tracking technologies for deployment. In this work, we explore optimization strategies for MP-QKD in a wavelength-division multiplexing scenario. The simulation results reveal that incorporation of multiple wavelengths not only leads to a direct increase in key rate but also enhances the pairing efficiency by employing our novel pairing strategies among different wavelengths. As a result, our work provides a new avenue for the future application and development of MP-QKD.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantum key distribution (QKD) [1] enables secure sharing of private keys between authorized users guranteed by the quantum mechanics [2,3]. By incorporating the one-time pad method, it can provide communication with information-theoretic security, even in the presence of a malicious eavesdropper. Over the past four decades, QKD has transitioned from theory to practical applications and has been advancing towards high-speed, miniaturized, and networked deployments [4–9].
Among the numerous developments in recent years, measurement-device-independent QKD (MDI-QKD) [10,11] has stood out for its notable advantages in terms of both security and practicality. With the detection part controlled by an untrusted third party, MDI-QKD can be immune to detection side-channel attacks and has been successfully verified in proof-of-principle demonstrations and practical systems [12–19]. However, the performance of BB84 and MDI-QKD protocols is limited by the fundamental rate-distance bound of QKD without quantum repeaters [20,21] (Pirandola-Laurenza-Ottaviani-Banchi bound, PLOB bound). Extending the transmission distance beyond 450 km presents a considerable challenge. Fortunately, in 2018, twin-field (TF) QKD [22–26] was introduced as a breakthrough solution to surpass this limit without a full-working repeater. To validate the feasibility of TF-QKD, numerous indoor and outdoor experimental demonstrations have been conducted [27–37], which utilized various techniques for both phase locking and phase tracking. Although researchers have reduced the requirements for phase calibration and stability [38,39], it has inevitably resulted in increased time and system complexity.
Recently, mode-pairing (MP) QKD [40], also known as asynchronous MDI-QKD [41], has been proposed as a means to alleviate the technological demands of TF-QKD [42–48]. Instead of pairing only adjacent time slots, MP-QKD recycles the unpaired events in MDI-QKD by actively postselects coincidence events within a suitable pairing interval ($l$). More specifically, when $l$ is set to 1, the relationship between the key rate ($R$) and channel transmittance ($\eta$) follows $R=O(\eta )$, and as $l$ tends to infinity, it becomes $R=O(\sqrt \eta )$. Although achieving an infinite $l$ is not feasible, increasing the value of $l$ can indeed enhance the performance of MP-QKD and, to a certain degree, surpass the PLOB bound. However, the maximum pairing interval is generally constrained by the number of emitted pulses within the laser’s coherent time. Nonetheless, improving the system frequency and laser performance presents demanding and costly challenges during system upgrades.
In this work, we propose a wavelength division multiplexing (WDM) scheme to improve the pairing efficiency of the MP-QKD. By employing this method, we can pair the effective events between different wavelengths within the pairing interval, and thus events that previously failed to be paired can be re-utilized across different wavelengths. A general formula for pairing an arbitrary number of wavelengths is presented. In addition, we validate the effectiveness of our proposed method through simulation and subsequently provide a summary and outlook.
2. WDM MP-QKD
In this section, we begin by reviewing the original procedures of MP-QKD. Subsequently, we introduce MP-QKD with WDM, focusing on its pairing strategy. Additionally, corresponding simulation formulas are provided.
2.1 MP-QKD
For time-bin phase-encoding MDI-QKD, a successful Bell state measurement requires the simultaneous clicks of counting events in two adjacent time slots. In contrast, TF-QKD only requires a single counting event. MP-QKD, through the utilization of a post-selection pairing strategy, enables the reutilization of single counting events that would be discarded in MDI-QKD. A schematic setup of MP-QKD is shown in Fig. 1(a) and the specific protocol flow is as follows.
1. State preparation: In $k$-th round ($k \in \{ 1,2,\ldots,N\}$), Alice (Bob) randomly prepares weak coherent pulses $\sqrt {\mu _k^a} {e^{i\phi _k^a}}$ ($\sqrt {\mu _k^b} {e^{i\phi _k^b}}$) with $\mu _k^a(\mu _k^b) \in \{ 0, u \}$ and $\phi _k^a(\phi _k^b) \in [0,2 \pi )$, and further sends to the detection side.
Fig. 1. (a)The schematic diagram of MP-QKD. (b) WDM MP-QKD with m channels. $D_m$ denotes the $m$-th detection component, which possesses an identical setup to $D_1$. (c) Schematic representation of the pairing strategy between different wavelength.
2. Measurement: Upon the pulses interfering at the beam splitter, Charlie records the clicking events of the two detectors: $R(L)=1$ to indicate clicking of the right (left) detector, $R(L)=0$ to indicate non-clicking of the right (left) detector, and ${C^k} = R \cap L$. A successful event is defined as ${C^k} = 1$.
3. Mode-pairing: After collecting enough data for $N$ rounds, Alice and Bob pairing two successful events with a specific strategy as follows [40]: First, a maximum pairing length, denoted as $l$, is predetermined by Alice and Bob based on the stability of the laser source and the system repetition rate. Let the index of the first slot of successful event be denoted as $i$, and the second as $j$. If the time interval between them is less than or equal to $l$, these two events are successfully paired as a coincidence event. Otherwise, the first successful event is discarded, and the attention is redirected to the time interval between the second and third events, and so on.
4. Basis sifting: Based on the intensities of the coincidence events (assuming the time slots are denoted as $i$ and $j$), Alice and Bob classify the data pairs in the $Z$ basis if the intensities $(\mu _i^a,\mu _j^a)$ are (0, $u$) or ($u$, 0), in the $X$ basis if the intensities are ($u$, $u$), or define it as ’0’ if the intensities are (0, 0). Subsequently, they publicly announce the post-defined bases and retain the $ZZ$ and $XX$, along with the ’0’ cases.
5. Key mapping: For the Z-basis, Alice classifies the key to ${\kappa ^a} = 0$ if $(\mu _i^a,\mu _j^a) = (0,u)$, and ${\kappa ^a} = 1$ if $(\mu _i^a,\mu _j^a) = (u,0)$; on the contrary, Bob classifies the key to ${\kappa ^b} = 0$ if $(\mu _i^b,\mu _j^b) = (u,0)$, and ${\kappa ^b} = 1$ if $(\mu _i^b,\mu _j^b) = (0,u)$; for the X-basis, the key of Alice is determined by the relative phase $(\phi _i^a - \phi _j^a) = {\theta ^a} + \pi {\kappa ^a}$ with ${\kappa ^a} = \left \lfloor {((\phi _i^a - \phi _j^a)/\pi )\bmod 2} \right \rfloor$, and ${\theta ^a} = (\phi _i^a - \phi _j^a)\bmod \pi$. Similarly, Bob determines his keys ${\kappa ^b}$ and alignment angle ${\theta ^b}$ in the same way, with the exception of performing a bit-flip when Charlie detects either $(L, R)$ or $(R, L)$ events. Then, they both announce the alignment angle, and keep the events when ${\theta ^a}={\theta ^b}$.
6. Parameter estimation and key generation: Slightly different from other protocols, MP-QKD should estimate the paring rate $r_p$ and probability of post-selected $ZZ$ basis $r_s$. Also, the fraction of the clicked signals ($q_{11}$) and single-photon phase error rate ($e_{11}^{XX}$) should be estimated. Finally, the secure keys can be generated after error correction and privacy amplification. Key rate of MP-QKD can be calculated as follows:
where $H(x) = - x{\log _2}(x) - (1 - x){\log _2}(1 - x)$ is the binary Shannon entropy function, ${E_{uu}^{ZZ}}$ is the quantum bit error rate in $ZZ$ basis, and $f$ is the error correction efficiency.2.2 WDM scheme
In this part, we will introduce the WDM MP-QKD protocol especially its paring strategy. WDM technology is widely employed in optical communications and network scenarios to enhance data transmission bandwidth and speed. Likewise, WDM technology can also be employed in QKD system to increase the key generation rate [49,50]. Different from previous studies, the application of WDM in this work not only leads to improvement in multiple wavelengths, but also allows for optimization of the pairing strategy in MP-QKD. This optimization enhances the pairing success probability and ultimately improves the key rate. Specific process of WDM MPQKD is as follows:
1. State preparation: Suppose Alice and Bob possess an MP-QKD system with $M$ distinct wavelengths. In $k$-th round, ($k \in \{ 1,2,\ldots,N\}$), Alice (Bob) randomly prepares weak coherent pulses $\sqrt {\mu _{k,m}^a} {e^{i\phi _{k,m}^a}}(\sqrt {\mu _{k,m}^b} {e^{i\phi _{k,m}^b}})$ with $m$-th wavelength ($m \in \{1, 2,\ldots, M\}$). Similarly, $\mu _{k,m}^a(\mu _{k,m}^b) \in \{ 0, u \}$ and $\phi _{k,m}^a(\phi _{k,m}^b) \in [0,2 \pi )$. As depicted in the Fig. 1(b), the pulses of all wavelengths are combined and sent to the Charlie through a wavelength-division multiplexer.
2. Measurement: At the measurement terminal, Charlie interferes the pulses from the same wavelengths and records the results of the $k$-th time slot and $m$-th wavelength as ${C^{k,m}}$.
3. Mode-pairing: Assume that the first two events are ${C^{i,m_1}}$ and ${C^{j,m_2}}$. As long as the time slot interval ($j-i$) between the two is smaller than the maximum pairing length $l$, regardless of which wavelength ($m_1$ and $m_2$) they originate from, Alice and Bob will successfully pair the two events as a coincidence. Otherwise, ${C^{i,m_1}}$ will be discarded and the attention is turn to the time interval between the ${C^{j,m_2}}$ and the next event, and so on. A simple schematic diagram is depicted in Fig. 1(c). Detailed pairing strategy are shown in Box 1.
4. Basis sifting and key mapping: As the encoding of different wavelengths is independent, the two steps remain almost the same as in the original MP-QKD. A slight difference arises during the X-basis key mapping process. Specifically, the relative phase between the two pulses from $m_1$ and $m_2$ channels can be denoted as $(\phi _{i,m_1}^a - \phi _{j,m_2}^a) = {\theta ^a} +{\delta _{m_1,m_2}} +\pi {\kappa ^a}$. ${\delta _{m_1,m_2}}$ is the phase difference introduced by different wavelengths traveling through the same optical fiber and this variation can be mitigated in practical applications through prior estimation [35,37].
5. Parameter estimation and key generation: This step remains identical to the original protocol. In the following section, we will mainly focus on the derivation of the parameter estimation formulas for WDM MP-QKD.
3. Simulation model for WDM MP-QKD
In this section, we will present the formulas for WDM MP-QKD, with a particular focus on the calculation of the efficiency for the new pairing strategy. In this work, we mainly consider the simulation model in the asymptotic case where Alice and Bob randomly emit pulses with intensity ${0,u}$, and the decoy-state part is also prepared for perfect parameter estimation. For the ease of presenting the simulation formulas, the emitted pulses by Alice (Bob) in $k$-th round is denoted as $\left |\sqrt {z_{k,m}^{a} u} \exp \left (i \phi _{k,m}^{a}\right )\right \rangle$ ($\left |\sqrt {z_{k,m}^{b} u} \exp \left (i \phi _{k,m}^{b}\right )\right \rangle$) with $z_{k,m}^{a}$ ($z_{k,m}^{b}$) $\in \{0,1\}$ and $\phi _{k,m}^{a}$($\phi _{k,m}^{b}$) is the random phase. In $k$-th round and $m$-th channel, the intensity setting is expressed as $z_{k,m}:=[z_{k,m}^a, z_{k,m}^b]$. Without loss of generality, we assume that Alice and Bob are equidistant from Charlie, and the channels follows the i.i.d assumption. By following the pairing strategy in Box 1, the $F_k$-th pulse from $m_{1k}$ channel and $R_k$-th pulse from $m_{2k}$ channel are paired. Define ${\tau _{{F_k},m_{1k},{R_k},m_{2k}}} = [\tau ^a_{{F_k},m_{1k},{R_k},m_{2k}},\tau ^b_{{F_k},m_{1k},{R_k},m_{2k}}]: = [z ^a_{{F_k},m_{1k}} \oplus z ^a_{{R_k},m_{2k}},z^b_{{F_k},m_{1k}} \oplus z^b_{{R_k},m_{2k}}]$, where $\oplus$ is the bit-wise modulo-2 addition. When ${\tau _{{F_k},m_{1k},{R_k},m_{2k}}} =[1,1]$, this pair is defined as Z-pair.
First, the successful detection probability of the each pulse-pair is
Specifically, in $k$-th round and $m$-th channel, the successful event probability $P(C_m^k =1|z_{k,m})$ can be expressed as
where $p_d$ is the dark count rate, and ${\eta _s}$ is the overall transmission efficiency including the detector.Moreover, the weak coherent pulses with random phase can be treat as a mix of the photon-number state, and the detection probability when Alice and Bob send $n^{k,m}$ photons ($n_A^{k,m}$ from Alice and $n_B^{k,m}$ from Bob) is
According to Ref. [40], the signal-pair ratio $r_s$ is
Besides, the single-photon ratio $q_{11}$ in the effective signal-pairs is
By using decoy-state method, we can perfectly estimate the single-photon yield and phase error rate by using the following formulas [40]:
The general expression for the expected pair number with $M$ wavelengths per pulse is
Detailed formula derivations can be found in Appendix A.
Finally, by employing Eq. (1) and the aforementioned formulas, the ultimate key rate can be computed.
4. Simulation results
In the previous section, we provided the simulation formulas and a general expression for the success rate of pairing. In this part, we will present corresponding simulation results to demonstrate the effectiveness of our proposed scheme. The simulation parameters are listed in the following Table 1.
Table 1. Simulation parameters. $\eta _d$ and $p_d$ are the efficiency and dark counting rate per pulse of the detectors. $\alpha$ is the standard fiber loss coefficient. $f$ is the error correction efficiency.
First, we undertake a direct comparison of the key rates following the integration of the WDM scheme with $l=1000$. In Fig. 2, it is evident that adopting the multiplexing approach significantly enhances the key rate. As the number of multiplexed channels increases, the key rate improvement becomes more pronounced, making it easier to surpass the PLOB bound. It is worth to note that regardless of the number of multiplexed channels, the maximum transmission distance remains unchanged. This is attributed to the dominant influence of long-distance noise.
Furthermore, to provide a clearer illustration of the advantages of WDM in MPQKD, we computed the ratio of the key rates ($M$=2, 3, 4, 5) to the original scenario($M$=1). As shown in Fig. 3, within the initial 200 km, the ratio is approximately equal to the number of channels M. This indicates that the enhancement at this term is from the expansion of wavelengths. Remarkably, a substantial enhancement follows, with the ratio increasing rapidly. For example, when $M$ equals 5, the ratio increases to 25 at a distance of 500 km, demonstrating an improvement by a factor of $M^2$. This highlights the efficacy of the WDM method in MPQKD, where it elevates the pairing success rate and enhances the ultimate key rate at long distance. In Appendix B, we discuss in detail the reasons behind the enhancements brought about by the WDM approach.
Lastly, we conducted a comparative analysis of the benefits derived from increasing the pairing interval $l$ and the number of multiplexed channels $M$. Increasing the pairing interval without compromising system performance implies a requirement for higher repetition rates or improved laser sources. Implementation of WDM demands lasers with different wavelengths which could be realized with optical frequency comb [51]. In Fig. 4, it is evident that for $M=1$, doubling or quintupling the pairing interval yields lower improvement in key rates compared to the enhancements achieved through adding multiplexing channels with the same factor. This phenomenon is more distinctly observable in the comparison chart of key rates (right-hand coordinate system). For instance, increasing $M$ by a factor of 5 ultimately results in an enhancement close to 25 times, whereas a 5-fold increase in $l$ only results in a 5 times improvement.
Fig. 4. Comparison of key rates among different $M$ and $l$. Left axis represents absolute key rate and PLOB bound is also plotted; right axis represents the relative key rate value with respect to $M=1$ and $l=1000$.
Hence, WDM scheme can lead to a more substantial enhancement of the key rate than the improvement of the pairing interval. This results provides valuable insights for future system upgrades in MPQKD.
In real-life scenarios, the finite-intensity decoy-state method are much more practical. As an illustrative addition, Fig. 5 depicts the key rate corresponding to Fig. 2, following the application of a three-intensity decoy-state method that incorporates statistical fluctuations [40,47]. The result in Fig. 5 suggests that the WDM scheme can enhance both the key rate performance and the maximum transmission distance when more complex practical factors are considered. Furthermore, when $M$ exceeds 2, the key rate can surpass the PLOB bound, demonstrating superior practical performance.
Fig. 5. Key rate of three-intensity decoy-state WDM MP-QKD with statistical fluctuation considered. Parameters taking from Ref. [47] are listed below. Total pulse number $N=10^{12}$; dark count rate $p_d=10^{-8}$; detection efficiency $\eta _d=70{\% }$; fiber loss coefficient $\alpha =0.2 dB/km$; error correction efficiency $f=1.1$; secure coefficient $\varepsilon =4.8\times 10^{-23}$; Z-basis misalignment error $e_d^Z=0.5{\% }$; X-basis misalignment error $e_d^X=5{\% }$. Key rate for $l =1000$ with $M$=1, 2, 3, 4, 5 are displayed, and also PLOB bound is presented for comparison.
5. Conclusion and outlook
In this work, we present a WDM-based MPQKD scheme that leverages an improved pairing strategy to enhance key rates. Our approach involves pairing events across different wavelengths, leading to gains in key rates not only through wavelength multiplexing but also by optimizing pairing efficiency. Furthermore, we derive a general expression for the arbitrary number of channel multiplexing. Simulation results demonstrate the substantial enhancement in key rates achieved by our scheme, particularly over longer distances. This offers valuable insights for future system upgrades in MPQKD.
In future research, particularly in the realm of experimental implementations, several areas merit attention. Firstly, when employing WDM scheme, it is essential that different wavelengths maintain a stable phase correlation in practical applications. One potential solution to achieve this stability is utilizing a frequency comb as the light source. Secondly, when light sources with different wavelengths are transmitted over the same fiber, additional phase variations can still be introduced. These variations can be measured and estimated during the experiment, and then compensated for in principle . Additionally, WDM may lead to channel crosstalk, introducing extra noise. These aspects represent potential challenges in experimental implementations that need to be validated.
Appendix A: Pairing efficiency of $M$ wavelengths
Here, we will focus on the derivation of the pairing efficiency $r_p$ which represents the pairing number of each pulse. According to the definition, it can be written as
where $n_{tot}$ is the total number of pairs during $N$ pulses.Firstly, we will revisit the derivation of $r_p$ in the original MPQKD protocol. According to the pairing strategy, two clicks should be found within the maximal pairing length $l$ to pair a successful event. For a given click event, the probability of encountering at least one additional click within the subsequent $l$ time slots is described by
As a result, it takes an average of $1 + \dfrac {1}{q}$ clicks for a successful event. Therefore, $n_{tot}$ can be given by
Consequently, $r_p$ can be rewritten as
In WDM scenarios where the wavelengths are expanded to $M$, the potential for click events increases correspondingly, scaling up by a factor of $M$. Also, this expansion necessitates a recalibration of the pairing pulses, modified to $Ml$. Consequently, $n_{tot}$ can be expressed as follows:
Finally, general expression for the expected pair number with $M$ wavelengths per pulse is
Appendix B: Analysis of improvement in WDM
In this part, we will discuss in detail the reasons behind the enhancements brought about by the WDM approach. The updated formula for $r_p$ is presented by Eq. (16), where we can see that $r_p$ is affected by the channel number $M$ and the successful detection probability of the each pulse-pair $p$. We remark that, when ${\eta _s}u \ll 1$, $p \approx {\eta _s}u$. Therefore, at close distances,
Consequently, $r_p \propto M$. Based on the key rate formula Eq. (1), it becomes evident that at close distances, the key rate increases $M$-fold, driven by an $M$-fold improvement in $r_p$.
Conversely, at longer distances, as $\eta _s$ tends to 0, $r_p$ can be reformulated as
To further clarify the relationship, we have added a new figure to illustrate the phenomenon.
It is clear from Fig. 6 that, when $p$ is relatively large, it follows a ratio of $M$ times. Conversely, for smaller $p$ values, it adheres to a multiple relationship of $M^2$.
Funding
National Natural Science Foundation of China (62101285, 12074194, 12104240, 62201276); Industrial Prospect and Key Core Technology Projects of Jiangsu Provincial Key Research and Development Program (BE2022071); Leading-Edge Technology Program of Natural Science Foundation of Jiangsu Province (BK20192001, BK20210582); Major Basic Research Project of the Natural Science Foundation of the Jiangsu Higher Education Institutions (21KJB140014, 22KJB510007).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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