Analysis for polarization entanglement distribution coexisting with classical light in fiber at telecom C-band

Abstract

Entanglement distribution between spatially distant nodes is crucial for advancing the development of quantum networks. Achieving large-scale, practical quantum networks by leveraging existing fiber infrastructures and commercial technologies can greatly enhance cost efficiency in real-world applications. As such, the simultaneous transmission of quantum entanglement and classical signals within a single optical fiber is becoming increasingly desirable, yet it remains challenging due to the substantial intensity disparities between single-photon-level quantum light and classical traffic. In this study, we theoretically describe and experimentally implement polarization entanglement distribution using a continuous wave pumped integrated AlGaAs Bragg reflection waveguide quantum source, coexisting with classical light in optical fiber at the telecom C-band. The characteristics of noise photons are analyzed and evaluated under different levels of classical light power. Using the BBM92 protocol for quantum key distribution, we demonstrate that temporal and spectral filtering during coincidence detection, combined with optimized source brightness, effectively influences the secure key rate and achievable distance of the system. Polarization drift is measured across various fiber paths, with 35-hour stability tests on fibers ranging up to 55 km in length. Our results can guide the initial design of optimal entanglement distribution networks that are compatible with fiber-optic communication systems.

Graphical abstract

Keywords

quantum entanglement distribution / coexistence / noise analysis / integrated photonics

1 Introduction

Quantum key distribution (QKD) is a theoretically secure method for generating secret keys between two spatially separated users [1]. Since its development in 2014 [2], various discrete-variable protocols have emerged, including entanglement-based methods (E91 [3] and BBM92 [4]), decoy states [5], measurement-device-independent approaches [6], and twin-field protocols [7]. Numerous studies have investigated QKD over free-space [8, 9], fiber [10–15], and water-based channels [16, 17]. Among these, entanglement-based protocols are particularly appealing, as they do not require quantum random number generators or additional active modulators and relinquish trust in the quantum source. On the receiving side, each participant measures incoming signals by randomly selecting between two mutually unbiased bases, implemented passively through probabilistic beamsplitters routing photons to the measurement bases. Attractively, entanglement-based protocols enable fully connected quantum network architectures without the need for trusted nodes, thereby supporting widespread connectivity [18–35]. Within such architectures, the nonlocal properties of distributed quantum states enable dedicated entanglement between any pair of users, even in the absence of a direct optical link. However, most of these demonstrations to date rely on dark fiber, wherein quantum and classical signals are transmitted over separate fibers. This is not a viable option under strict operational expenditure constraints or in scenarios where fiber resources are limited. Leveraging existing fiber infrastructures and conventional technologies is essential for the efficient, cost-effective deployment of quantum networks.

While showing great promise, the stark contrast in brightness between classical and quantum signals presents a significant challenge for their simultaneous transmission of QKD and optical communication in standard telecom fibers. Townsend proposed the first coexistence scheme for QKD and conventional data over installed fiber using wavelength-division multiplexing (WDM) techniques [36]. Subsequently, a series of quantum-classical coexistence studies have been conducted [37–43]. Due to the spectral overlap of noise photons generated by classical light with quantum signals, quantum and classical channels are typically allocated to widely separated frequencies. For instance, by strategically allocating quantum signals to the telecom O-band (1260−1360 nm) and classical signals to the C-band (1530−1565 nm), researchers have demonstrated the real-world coexistence of QKD systems with conventional internet traffic [44, 45], maintaining autonomous operation for over two weeks [44]. Given that the O-band exhibits marginally higher attenuation than the C-band (approximately 0.1−0.2 dB/km difference), this configuration needs to balance the O-band’s attenuation penalty against the achievable key rate metrics [45]. Recently, the coexistence of both quantum and classical signals within the telecom C-band has been demonstrated, using attenuated lasers as proxies for quantum signals [46, 47]. Unlike attenuated lasers, the inherent spectral and temporal correlations in entangled photons can contribute to the differentiation between true coincidences and uncorrelated background counts. Thus, achieving coexistence between entanglement distribution and classical light in the telecom C-band represents a promising avenue for quantum communication. Towards this end, energy−time entanglement distribution coexisting with 20 Gbps fiberoptic communication over a distance of 40 km within the telecom C-band has been recently demonstrated [48]. The time-bin qubit is one of the leading contenders for fiber networks because this encoding does not suffer from random polarization rotation in fibers [49]. Compared to time-bin encoding, polarization encoding provides more straightforward control and analysis, placing less overhead on maintaining interferometric phase stability. While environmental factors such as temperature fluctuations and mechanical stress have strong impacts on the polarization drift [50], advances in active stabilization techniques now enable robust polarization entanglement distribution, with field tests demonstrating stable operation exceeding tens of hours in deployed fiber networks [51, 52]. The polarization state manipulation devices employed are fully interchangeable with the cost-effective polarization compensation modules used in commercial optical fiber communications without the need to customize additional high-bandwidth light sources or detectors. However, the coexistence of both polarization entanglement distribution and classical light within the telecom C-band has not yet been fully explored. The key technical hurdles include the development of scalable quantum hardware, compatibility with existing infrastructure, and suppression of noise, environmental fluctuations, and channel crosstalk.

In this study, we investigate a coexistence scheme for polarization entanglement distribution with an integrated source alongside classical light using WDM within the telecom C-band through spectral and temporal filtering. Compatibility with legacy infrastructure is achieved through optical add-drop multiplexers, with full adherence to International Telecommunication Union-Telecommunication Standardization sector (ITU-T) channel standards. We provide a theoretical description and experimental demonstration of the noise factors influencing entanglement distribution distance and secure key rate (SKR) under varying classical light power. To provide a benchmark for field experiments, we perform polarization drift measurements across various fiber paths. An analytical framework is developed to quantitatively evaluate the impact of fiber-induced polarization drift on polarization entanglement distribution systems. Long-term stability is assessed over a 35-hour period through sequential testing of fiber lengths up to 55 km. Our results indicate that the quantum correlations of photon pairs enhance the robustness of quantum channels and demonstrate the feasibility constructing a noise-robust and resource-efficient quantum internet within a shared multiplexing environment. While this study focuses on co-transmission between two users, the inherent compatibility of its hardware and protocols allows for a straightforward extension to multi-user scenarios [53–55].

2 Model for CW-pumped QKD coexisting with classical light

2.1 CW-pumped entangled source

In entanglement-based QKD utilizing the BBM92 protocol [4], polarization and time-bin encoding are commonly used. The system is divided into two parts: a central transmitter (responsible for entangled-state preparation) and two spatially separated receivers (Alice and Bob), each randomly choosing either the X or Z basis for projective measurement. They retain only the results with matching bases and proceed with post-processing [56, 57]. The state emitted by the entangled source is independent of the measurement bases chosen by the receivers, providing source-independent security [4, 9, 58]. Here, we focus on polarization entanglement generated via the spontaneous parametric down-conversion (SPDC) process, which aligns with our experimental configuration [32, 57, 59, 60]. The interaction Hamiltonian can be expressed as [61, 62]

(1){\hat{H}}_I={\displaystyle \frac{C}{\sqrt{2}}}{\displaystyle \sum _{k=1}^N\sqrt{{\lambda }_k}(A{}_{k,H}^{†}B{}_{k,V}^{†}+A{}_{k,V}^{†}B{}_{k,H}^{†})+H.c.,}

where C is a constant that depends on the pump intensity and the effective nonlinear coefficient. The creation operator A{}_{k,H(V)}^{†}, B{}_{k,H(V)}^{†} represents a signal (idler) photon with H(V) polarization in the k-th spectral mode. The coefficient {\lambda }_k gives the probability for the entangled state to populate the k-th spectral mode. The state emitted from the source can be written as [61]

(2)\begin{array}{rl}|\mathrm{\Psi }⟩=& {\otimes }_{k=1}^N\exp [\frac{-\mathrm{i}}{\hslash }\frac{C\sqrt{{\lambda }_k}}{\sqrt{2}}(A{}_{k,H}^{†}B{}_{k,V}^{†}+A{}_{k,V}^{†}B{}_{k,H}^{†})]|0⟩\\ =& {\otimes }_{k=1}^N|{\psi }_k⟩,\end{array}

where {\otimes }_{k=1}^N denotes the tensor product of modes, and |{\psi }_k⟩ represents the Fock state in the k-th spectral mode, expressed as [63]

(3)|{\psi }_k⟩={\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}}^{-2}x{\displaystyle \sum _{n_k=0}^{\mathrm{\infty }}\sqrt{n_k+1}{\tanh }^{n_k}x|{\mathrm{\Phi }}_{n_k}⟩,}

where x=\frac{C\sqrt{{\lambda }_k}}{\sqrt{2}\hslash }, and |{\mathrm{\Phi }}_{n_k}⟩ is the state of an n_k-photon pair in the k-th mode, given by

(4)\begin{array}{rl}|{\mathrm{\Phi }}_{n_k}⟩=& \frac{1}{\sqrt{n_k+1}}\sum _{m_k=0}^{n_k}{|n_k-m_k⟩}_{H,A}{|m_k⟩}_{V,A}\\ & \cdot {|n_k-m_k⟩}_{V,B}{|m_k⟩}_{H,B},\end{array}

where {|n_k-m_k⟩}_{H,A}{|m_k⟩}_{V,A}{|n_k-m_k⟩}_{V,B}{|m_k⟩}_{H,B} represents a tensor product of photon number states, with n_k-m_k photons in modes A_{k,H} and B_{k,V}, and m_k photons in modes A_{k,V} and B_{k,H}. The probability of creating n_k entangled photon pairs in the k-th mode is

(5)P(n_k)={\displaystyle \frac{(n_k+1){\sinh }^{2n_k}x}{(1+{\sinh }^{2}x{)}^{n_k+2}}}.

The expected photon pair number is {\xi }_k=2{\sinh }^{2}x, representing the average number of photon pairs generated in the k-th mode [61].

In current QKD systems, threshold detectors are commonly used. Although they cannot determine the number of incoming photons, they provide high practicality. We define {\eta }_{SPD,A} and {\eta }_{SPD,B} as the detector efficiencies, whereas {\eta }_{loss,A} and {\eta }_{loss,B} represent the probabilities of photon arrival, accounting for all system losses before detection. Thereby {\eta }_A={\eta }_{SPD,A}{\eta }_{loss,A} and {\eta }_B={\eta }_{SPD,B}{\eta }_{loss,B} represent the total efficiencies for the users. For a CW-pumped source, let Y(n_1,...,n_k,...,n_N,t_{cc}) denote the yield of an n-photon pair (n=\sum _k{n}_k) in the N modes. We define t_{cc} as the chosen coincidence window, and use n_k to represent n_k-photon pair in k-th mode, which originates from two distinct contributions: background noise and entangled photons. Considering standard non-photon-number-resolving detectors, an avalanche event is registered if at least one photon triggers the response. Under the assumption that background counts are statistically independent of true signals, then Y(n_1,...,n_k,...,n_N,t_{cc}) can be expressed as

(6)\begin{array}{rl}& Y(n_1,...,n_k,...,n_N,t_{cc})\\ & ={\eta }_{t_{cc}}[1-(1-{\eta }_A{)}^{\sum _k{n}_k}][1-(1-{\eta }_B{)}^{\sum _k{n}_k}]\\ & +(D_A+N_A)(1-{\eta }_A{)}^{\sum _k{n}_k}[1-(1-{\eta }_B{)}^{\sum _k{n}_k}]\\ & +(D_B+N_B)[1-(1-{\eta }_A{)}^{\sum _k{n}_k}](1-{\eta }_B{)}^{\sum _k{n}_k}\\ & +(D_A+N_A)(D_B+N_B)(1-{\eta }_A{)}^{\sum _k{n}_k}(1-{\eta }_B{)}^{\sum _k{n}_k},\end{array}

where D_{A(B)} denotes the dark count rate per coincidence window of Alice’s (Bob’s) detectors and N_{A(B)} represents the background noise count rate per coincidence window on the two users’ sides [see Eqs. (13) and (14)]. We use D_{A0(B0)} to denote the dark count rate per second, with N_{A0(B0)} defined analogously. Thus, we have: D_{A(B)}=D_{A0(B0)}{t}_{cc} and N_{A(B)}=N_{A0(B0)}{t}_{cc}. To simplify the subsequent formulas, we use N_{A(B),tol} to represent the total noise count rate in Alice’s (Bob’s) detection system, where N_{A(B),tol}=D_{A(B)}+N_{A(B)}, and {\eta }_{t_{cc}}=\mathrm{e}\mathrm{r}\mathrm{f}[\sqrt{\mathrm{l}\mathrm{n}2}{t}_{cc}/t_{\mathrm{\Delta }}] is coincidence-window dependent detection efficiency. t_{\mathrm{\Delta }} is the total timing imprecision of the system, which can be read from the full width at half maximum of the temporal distribution, and t_{cc} is the chosen coincidence window. Meanwhile, the dead time-dependent loss needs to be considered in {\eta }_A and {\eta }_B when the count rates are high. In this case, the detector efficiency should be corrected as {\eta }_{SPD,A(B)}/[1+{\tau }_{dead}(B{\eta }_{loss,A(B)}+N_{A(B)})/d], where B is source brightness, d is the number of detectors used by each user and {\tau }_{dead} is the detector’s dead time. We use d=2 in the simulation, which is consistent with our experimental configuration. Eq. (6) decomposes the coincidence statistics into three distinct components. The first term characterizes true coincidences, where both detectors simultaneously register entangled photon pairs from the source. The second and third terms account for mixed events, comprising one genuine photon detection alongside an independent background count. The final term quantifies contributions from simultaneous noise counts in both detectors. Accordingly, the overall gain is given by

(7)Q_{\xi }=\sum _{n_1,\cdots ,n_k,\cdots ,n_N}Y\left(n_1,\cdots ,n_k,\cdots ,n_N,t_{cc}\right)\prod _{k=1}^N{P}_k\left(n_k\right).

Considering that the JSA exhibits strong spectral correlation in a CW-pumped soures, the photon number n_k is thus contributed by all combinations of spectral modes, when the modes number N\to \mathrm{\infty }, the overall gain becomes

(8)\begin{array}{rl}{Q}_{\xi }=& {\eta }_{t_{cc}}\left(1-{\mathrm{e}}^{-{\eta }_A\xi }-{\mathrm{e}}^{-{\eta }_B\xi }+{\mathrm{e}}^{-{\eta }_A\xi -{\eta }_B\xi +{\eta }_A\xi {\eta }_B\xi }\right)\\ & +\left({\mathrm{e}}^{-{\eta }_A\xi }-{\mathrm{e}}^{-{\eta }_A\xi -{\eta }_B\xi +{\eta }_A\xi {\eta }_B\xi }\right)N_{A,tol}\\ & +\left({\mathrm{e}}^{-{\eta }_B\xi }-{\mathrm{e}}^{-{\eta }_A\xi -{\eta }_B\xi +{\eta }_A\xi {\eta }_B\xi }\right)N_{B,tol}\\ & +{\mathrm{e}}^{-{\eta }_A\xi -{\eta }_B\xi +{\eta }_A\xi {\eta }_B\xi }{N}_{A,tol}{N}_{B,tol},\end{array}

where \xi denotes the average number of photons per time unit, expressed as \xi =B{t}_{cc}, with B represents the source brightness. Finally, the overall quantum bit error rate (QBER) is given by (see Appendix A for details)

(9)\begin{array}{rl}& E_{\xi }=e_0\\ & -\frac{2\left[e_0-{\mathrm{e}}^{pol}-\frac{1-2{\mathrm{e}}^{pol}}{4}\left(N_{A,tol}+N_{B,tol}-N_{A,tol}{N}_{B,tol}\right)\right]}{Q_{\xi }}\\ & \times {\eta }_{t_{cc}}\left[{\mathrm{e}}^{-\frac{\xi }{2}\left({\eta }_A+{\eta }_B-{\eta }_A{\eta }_B\right)}-{\mathrm{e}}^{-\frac{\xi }{2}\left({\eta }_A+{\eta }_B\right)}\right],\end{array}

where e_0 is the error probability which is equal to 1/2, and {\mathrm{e}}^{pol} is the intrinsic detector error probability which is related to the experimental system. Notably, in reality, in reality, the error probability should include both the intrinsic error of the detector and the error caused by the polarization drift of photons in the optical fiber. In Appendix B, we address this under more general conditions and use the Jones matrix formalism to quantitatively evaluate the polarization-dependent error. This error vanishes when the Jones matrix is an identity matrix, or when the squares of its off-diagonal elements equal 1, corresponding to the cases of no polarization inversion or complete polarization inversion, respectively.

For the BBM92 protocol, the SKR can be expressed as [58]

(10)R\ge {\displaystyle \frac{1}{2}}{\displaystyle \frac{Q_{\xi }}{t_{cc}}}[1-f(E_{bit})H_2(E_{bit})-H_2(E_{ph})],

where E_{bit} and E_{ph} are the bit and phase error rates, which are equivalent due to the symmetry of X and Z bases measurements. f(E_{bit}) is the error-correction efficiency, and H_2(x) is the binary entropy function

(11)H_2(x)=-x{\log }_2(x)-(1-x){\log }_2(1-x).

2.2 Co-propagating noise

In hybrid quantum-classical communication systems, high-power classical light adversely impacts QKD performance. As shown in Table 1, we summarize the types of noise in quantum/classical coexistence system along with their corresponding physical origin and characteristics. The primary sources of noise include crosstalk from classical light when WDM provides insufficient isolation and nonlinear noise, which primarily consists of spontaneous Raman scattering (SpRS), stimulated Raman scattering (SRS), Brillouin scattering, and four-wave mixing (FWM) [64]. In the case of crosstalk, noise depends on the relative strength of classical channels and the isolation provided by filters. The detection probability per coincidence window is given by

(12)N_{cross,A(B)}={\displaystyle \frac{P_{class}{\eta }_{cross,A(B)}{\eta }_{iso,A(B)}{\eta }_{SPD,A(B)}}{h{\nu }_{cross}}}{t}_{cc},

where P_{class} denotes the classical optical power, {\eta }_{cross,A(B)} represents the optical path component loss, {\eta }_{iso,A(B)} denotes the isolation degree of WDM, h is Planck constant, and {\nu }_{cross} is the classical optical frequency. In terms of scattering noise, depending on whether a phonon is excited or de-excited, Stokes or anti-Stokes emissions occur, with photoluminescence appearing at energies lower or higher than that of the excitation photons. Scattering from acoustic phonon (Brillouin scattering) is negligible, as the maximum frequency shift of scattered photons is relatively small (approximately 10 GHz in the backward direction), preventing interference with adjacent channels in telecommunication systems operating at 100 GHz spacing. In contrast, Raman scattering occurs due to inelastic interactions between optical photons and pump photons. The spontaneous Raman process can convert photons from the classical channel into a broad wavelength band, resulting in unpolarized noise photons that overlap in-band with the quantum signal along the fiber [68], making spectral filtering ineffective. As the intensity of an incident light increases, the scattered Raman field is enhanced, allowing an initially scattered Stokes photon to promote further scattering of additional incident photons. This process, where the Stokes field grows exponentially, is known as SRS. Since SRS cannot occur from a threshold significantly above the total classical optical power in telecommunications [40], thus the primary noise source considered here is SpRS. Co- and counter-propagating SpRS noise generated by a single classical light source in the fiber can be obtained [40, 64]:

(13)N_{ram,A(B)}={\displaystyle \frac{P_{ram,A(B)}{\eta }_{ram,A(B)}{\eta }_{SPD,A(B)}}{h{\nu }_{A(B)}}}{t}_{cc},

where

(14)P_{ram,A(B)}=P_{class}{e}^{-\alpha L}L\beta ({\lambda }_c,{\lambda }_{q_{A(B)}})\mathrm{\Delta }\lambda ,

denotes the Raman scattering power, \alpha is the attenuation coefficient, L represents the fiber length, \mathrm{\Delta }\lambda is the channel bandwidth, {\eta }_{ram,A(B)} is the loss from the fiber outlet to the detector, and \beta ({\lambda }_c,{\lambda }_{q_{A(B)}}) represents the Raman cross section of classical light with wavelength {\lambda }_c at the wavelengths of {\lambda }_{qA(B)} of quantum light, which can be inferred from Raman spectroscopy, and {\lambda }_{qA(B)} represents the quantum light wavelength that is sent to Alice (Bob). Unlike scattering noise, FWM is a parametric process where four photons interact through third-order nonlinearity. This interaction occurs if at least two photons propagate simultaneously within the fiber. The generation process can be classified into degenerate and non-degenerate cases, depending on whether the generated photons have the same frequency. The most detrimental situation arises when the quantum channel overlaps with the two newly generated frequencies, 2{\nu }_A-{\nu }_B and 2{\nu }_B-{\nu }_A, originating from two co-propagating input frequency components, {\nu }_A and {\nu }_B (where {\nu }_B>{\nu }_A). Generally, the noise from spontaneous FWM can be estimated by \mathrm{\Gamma }{P}_{class}L, where \mathrm{\Gamma } is the nonlinear parameter [69]. In most configurations, this value is negligible. Overall, the influence of FWM can be minimized by selecting an appropriate frequency difference between two co-propagating channels, thereby preventing the generation of FWM frequency products within the quantum channels [64].

3 Simulation

In this section, we present numerical simulations based on the preceding analysis. The simulation results indicate that the presence of classical light has a serious effect on maximum distance due to the increased accidental coincidences. Minimizing the background noise is essential for successful protocol execution in our quantum communication system. Unlike dark counts, which are intrinsic detector noise determined by detectors’ structure and operating conditions, producing uniformly distributed random errors, coherent noise from classical communication lasers exhibits wavelength-dependent power coupling characteristics. The detrimental impact of the coherent noise on the key generation can be reduced through spectral and temporal filtering techniques, or by designing advanced protocols such as leveraging the advantages of high-dimensional entanglement [32, 33, 70–72]. The simulation parameters, provided in Table 2, are feasible with current technology. Figure 1 illustrates the attainable maximal secure key rates as a function of source brightness under varying classical light powers. This corresponds to realistic scenarios, where channel and device parameters are fixed, while brightness and coincidence windows can be adjusted within specific limits. We assume a relatively weak SPDC process with and an overall quantum channel loss of 9 dB per photon, and the transmission distance is 30 km. We set E_{bit}=E_{ph}=E_{\xi } and f(E_{bit})=1.1 [32] during the simulation. For each curve, the coincidence window is first optimized using a genetic algorithm to maximize the SKR and is then kept constant. Given typical parameters of InGaAs detectors and superconducting nanowire single photon detectors (SNSPDs) in the telecom band, the numerical simulation in Fig. 1(a) confirms up to a two-order magnitude improvement in key rate for SNSPDs. The identified optimal brightness is fully compatible with existing high-performance sources [73, 74]. At lower brightness, the system signal is mainly composed of noise by dark counts. At higher brightness, both detector saturation and dead time lead to a decrease in the security key rate. It is observed that the optimal brightness shifts toward larger values as noise photon contributions increase. Furthermore, when the InGaAs detectors are used, the corresponding optimal brightness shift toward a lower value given the same noise photon contribution. It is mainly due to the increased dead time, making the detector saturation effect obvious. The cutoff at high brightness can be attributed to both the increased multi-photon events probability and detector saturation effect. Figure 1(b) illustrates how the optimal coincidence window (at optimal brightness) varies with the timing imprecision under different classical light powers. The observed trends reveal that under high classical light power, suppressing classical light-induced noise — which is more critical to QKD performance than enhancing the coincidence counts — forces a reduction in the optimal coincidence window t_{cc} for a given t_{\mathrm{\Delta }}, and likewise reduces the ratio t_{cc}/t_{\mathrm{\Delta }} as timing imprecision worsens at fixed power. To further illustrate the impact of classical light on QKD limits, Fig. 2 shows the secure key rate as a function of link loss, with both brightness B and coincidence windows optimized for each point. It is noteworthy that the secure key rate decreases significantly with increasing classical light power, and this reduction is more pronounced with SNSPDs. This occurs because, in the absence of classical light, the primary source of noise is the dark counts from detectors. For SNSPDs, the low dark count rate results in a high signal-to-noise ratio (SNR), allowing the system to tolerate high losses without classical light. In the presence of classical light, induced noise photons far exceed dark counts, making classical light noise the dominant factor and causing a sharp decline in maximum transmission distance. Simultaneously, the higher dead time results in a more rapid decline in the key rate of InGaAs detectors compared to SNSPDs, particularly at short transmission distances. As the transmission distance increases, the effect of dead time reduces, and the downward trend becomes relatively flat for the case of using InGaAs detectors.

4 Experimental results

To demonstrate the validity of our theoretical model, we implemented a coexistence scheme for polarization entanglement distribution and classical light using WDM in the telecom C-band. Figure 3 shows the elements of the experimental setup, containing three parts: photon-pair source, classical-quantum link, and polarization analysis module (PAM). Figure 3(a) illustrates the conceptual scheme for directly producing polarization-entangled photons from an AlGaAs Bragg reflection waveguide (BRW) chip through degenerate Type-II SPDC. In this configuration, a maximally entangled Bell state |{\mathrm{\Psi }}^{+}⟩=\frac{1}{\sqrt{2}}(|H{⟩}_{{\omega }_{s,A}}|V{⟩}_{{\omega }_{i,B}}+|V{⟩}_{{\omega }_{s,A}}|H{⟩}_{{\omega }_{i,B}}) is generated. The BRW comprises stacked AlGaAs layers with varying aluminum concentrations and has a length of 9.5 mm [75–77]. Thanks to the platform’s dispersion and nonlinear properties, the monolithic source can maintain high-fidelity polarization entanglement with high brightness over a broad bandwidth [21, 32]. For a detailed description of the source’s working principle and performance, we direct the reader to Ref. [60]. Figures 3(b)−(d) show the schematic of our experimental setup, consisting of the entanglement source, quantum-classical co-transmission link, and polarization analysis module. In the source section, an 8-mW CW pump laser with a wavelength centered at 780.068 nm is polarized and coupled into the BRW. The total insertion loss of the chip is about 10 dB, including input-output coupling loss and propagation loss in the chip for the TIR modes. The loss is primarily attributed to three factors: residual impurities from material growth and doping, defects introduced during the etching process, and significant mode-field mismatch between the waveguide and the single-mode fiber. To effectively reduce the loss, several strategies can be implemented: optimizing material growth parameters to lower residual impurity concentrations, applying a passivation layer (e.g., Al₂O₃ or SiO₂) to the waveguide surface to mitigate oxidation-induced absorption loss, and tailoring the waveguide width for better mode-field alignment with the single-mode tapered lens fiber. The generated polarization-entangled photons exhibit anti-correlation in frequency due to energy conservation during the SPDC process. At the output, the pump and most photoluminescence are suppressed by long pass filters (LPFs), and entangled photons are coupled into coarse wavelength-division multiplexing (CWDM) and distributed to two users, Alice and Bob. Alice uses a waveshaper for wavelength selection and filtering, while Bob uses a 200 GHz dense wavelength division multiplexer (DWDM) for filtering. The waveshaper performs three key functions: (i) reduces crosstalk from the classical signal; (ii) mitigates degradation due to noise photons; (iii) applies tunable bandpass filters on the signal spectra for bandwidth dependent measurements. Due to energy conservation during the SPDC process, the central wavelengths of the filters exhibit strong correlation. Therefore, the anticorrelation in frequency can be used to distill the signal photons from noise photons. Asymmetric filtering induces false coincidences from either noise-noise photons or noise-signal photons, thereby leading to accidental coincidence in experiments. Since more photons would pass through at the low-loss side, a variable optical attenuator (VOA) before Bob’s PAM is set with 5 dB loss to balance the losses and also prevent the detectors from saturating. In the experiment, an external noise source, as shown in Fig. 3(d), is introduced by multiplexing a 1540.56 nm [ITU-T C46] classical laser with signal and idler photons through separate 3 km fibers. We used a linearly polarized CW classical light (specifically H-polarization), which matches the H/V basis used for the quantum signals. On the receiving side, the classical and quantum signals are separated by filters, and the quantum signals are measured with random basis selection. Due to the lack of SPDs, we manually rotate the waveplates onto Z or X basis in each path and record the detection events using a time tagging unit. The SPDs (A1, A2, and B1, B2) are set with 10% detection efficiency, 0.6 kHz (0.8 kHz for B2) dark count rates, and 10 μs dead time. From these time-tag records, we can extract both single counts and coincidence counts.

Before performing entanglement distribution, we evaluate the performance of the source. The single photon counts in signal and idler channels at low pump power can be written as

(15)\begin{array}{rl}& N_A^m=S_A^t+D_{A_0}+\zeta P_{pump}^{\gamma }{\eta }_A,\\ & N_B^m=S_B^t+D_{B_0}+\zeta P_{pump}^{\gamma }{\eta }_B,\end{array}

where

(16)S_A^t=B{\eta }_A,S_B^t=B{\eta }_B,

and \zeta is the photoluminescence generation efficiency, which is mainly caused by impurities in the materials [78], and P_{pump} is the pump power. Here, we take into account the influence of dead time in a general manner, so the measured single photon rate can be expressed as

(17)S_A^m=N_A^m{\eta }_A^{\ast },S_B^m=N_A^m{\eta }_B^{\ast },

where {\eta }_{A(B)}^{\ast }=1/[1+{\tau }_{dead}{N}_{A(B)}^m/d], and d=1 is the number of detectors used by each user. The measured coincidence counts (C{C}^m) and accidental coincidence counts (C{C}^{acc}) can be calculated as

(18)C{C}^m={\eta }_{t_{cc}}B{\eta }_A{\eta }_A^{\ast }{\eta }_B{\eta }_B^{\ast }+C{C}^{acc},

(19)C{C}^{acc}=S_A^m{S}_B^m{t}_{cc}.

t_{cc}=0.5\mathrm{n}\mathrm{s} is the chosen coincidence window. Figure 4(a) shows the measured and simulated count rates of signal and idler photons under different pump powers. In order to get a clear characterization of photoluminescence signals, we used a transverse magnetic (TM)-polarized pump light to avoid the generation of biphotons. By fitting the measured data, the photoluminescence parameters are \zeta =0.1625\mathrm{M}\mathrm{H}\mathrm{z}/\mathrm{m}\mathrm{W}/\mathrm{n}\mathrm{m} and \gamma =1, respectively. Ideally, S_A^m(S_B^m) depends linearly on the pump power. However, as the pump power increases, the detector tends to saturate, causing the rate to deviate from the linear curve. Figure 4(b) shows the coincidences and the coincidence-to-accidental ratio (CAR) under different pump power, which shows that the theoretical model agrees well with measured results. From which, we can deduce that the source brightness is 0.25\mathrm{M}\mathrm{H}\mathrm{z}/(\mathrm{m}\mathrm{W}\cdot \mathrm{n}\mathrm{m}).

From the above theoretical analysis, we note that coexistence of classical and quantum signals in the same fiber can lead to some problems that may affect the performance of entanglement distribution. The main noise photons are from classical light crosstalk when the used DWDM has insufficient isolation and Raman scattering photons (RSPs) which occurs due to the inelastic photon−phonon interaction [64]. In order to obtain the Raman cross section for different wavelength, we measured the Raman scattering spectrum by launching an 8-mW classical light centered at 1540.56 nm (centered in the data transmission C-band) into the single mode fiber. The Raman cross section in a single mode fiber for an arbitrary pump excitation wavelength within the telecom band can be deduced from the measured results for a specific wavelength [32, 79]. This also implies that our theoretical model can be applied to scenarios involving classical lights with different wavelengths, intensities and propagation directions. The only modification required is to replace N_{A(B),tol} in Eq. (9) with the sum of all relevant noise counts. The measured Raman scattering cross section is shown in Fig. 5(a) with the signal (C16) and idler (C26) photons wavelengths marked. We first measure the single count rates at quantum channels with co-propagating classical light to characterize the property of noise photons from classical channel. In this case, the entangled photons are not included in the fiber. To specifically distinguish between crosstalk and RSPs, we measured the polarization-dependent photons to quantify the amount of crosstalk from polarized classical light. The RSPs can be obtained by detecting the contributions of orthogonal polarization at quantum receivers. Figure 5(b) gives the results of crosstalk and RSPs in C16 as a function of classical light power, which shows that the measured noise count rate increases linearly, which is consistent with the theoretical calculations.

Based on the above results, we take co-propagation as an example, recording 120 s coincidence count measurements across all tested scenarios. The entangled source is set to yield an average brightness of 0.96 MHz with corresponding measured coincidence rates of 10 Hz (cps) when using 200 GHz spectral filters, achieved with an on-chip pump power of 0.48 mW. Figure 6(a) illustrates the averaged QBER in both the X and Z bases. It is evident that as the coincidence window is broadened, the averaged QBER gradually increases due to the rising accidental counts. The growth trend continues to expand with the increase of classical light power, even exceeding the threshold E_{max}\sim 0.102, beyond which secure key distillation becomes infeasible. Unlike QBER, which assesses the quality of entanglement distribution, the distillable SKR incorporates both quality and quantity into a single formula. They depend on flux in significantly different ways [80]. As depicted in Fig. 6(b), the SKR reaches an optimal point as the window widens, and the optimal coincidence window decreases in the case of high classical power. In Fig. 7, we investigated the influence of channel bandwidth on QKD performance under varying classical optical powers. Utilizing the experimental configuration shown in Fig. 3, we dynamically tuned the signal channel bandwidth via a waveshaper, while employing a fixed 200 GHz DWDM (1.6 nm bandwidth, 80% transmission efficiency) for idler photon filtering. The source brightness B is kept as 2\times {10}^6(cps) with the coincidence window is optimized for each point. As shown in Fig. 7(a), when the bandwidth is less than 0.2 nm, the QBER rapidly declines with the sharp rise in coincidence counts. Beyond that, the QBER is almost unchanged when the signal channel bandwidth is below 200 GHz, because the coincidences and accidental coincidences (ACCs) are simultaneously increased as the bandwidth widens. Figure 7(b) shows that the SKR is an increasing function of the bandwidth. Further increasing the channel bandwidth beyond 200 GHz results in a rapid rise in accidental coincidences but without an increase in true coincidences, which will significantly increase QBER while simultaneously reducing the SKR. Therefore, the bandwidth optimization is crucial for the highest obtainable key rate in realistic applications. Meanwhile, the optimal setting of source brightness is key for the resulting final key rate, which can be adjusted up to a certain level by varying the laser pump power of the source. Current state-of-the-art sources with laser powers of several hundred milliwatts can readily achieve brightness values of up to {10}^{10}cps [74]. Finally, we dissect the impact of source brightness and classical light power on constraining the SKR (Fig. 8). The coincidence window is optimized for each scenario. The transmission loss across each channel is maintained at 15 dB, and the source brightness is kept at a low level to avoid detector saturation. In our experimental setup, with a classical power of 20 μW, equivalent to −17 dBm at the receiver — matching the minimal requisite power for a 20 Gbps classical signal [48] — we achieved an SKR of 1.85 bps and a QBER of 0.076. Although the SPDs used in our experiment has a relatively high dark count rate and a long dead time, which currently constrain the SKR, significant improvements in system robustness and SKR are achievable. These enhancements can be realized through the adoption of higher-performance detectors, along with optimizations to the source chip design and the use of advanced fabrication materials.

5 Polarization drift and stability of our system

To establish a benchmark for field experiments, we conduct polarization drift measurements across multiple fiber-optic paths, assessing long-term stability through 35-hour continuous testing of fiber segments up to 55 km in length. Polarization variations in deployed fibers exhibit complex and dynamic behavior, influenced by both the fiber length and external environmental conditions along the channel. A quantitative assessment of the relationship between polarization variations and the QBER is provided in Appendix B. In reality, environmental perturbations, representing an ensemble of uncontrolled variables in field deployment scenarios, resulting in unpredictable effects on polarization drift dynamics. For instance, the installed fibers can be classified into buried and aerial fibers [81]. Since aerial fibers are exposed to the open air, wind-induced vibrations and temperature fluctuations cause more severe polarization variations compared to buried fibers [50, 82]. On the other hand, field experiments employing underground fibers may show higher polarization stability relative to laboratory environments, primarily due to the inherent thermal stability [83]. Here we benchmark buried fiber experiments by characterizing the fibers’ drift rates in both X and Z bases over dynamically selectable paths (22 km to 55 km) under passive conditions to determine the necessary realignment intervals. The quantum signals coexist with 10-μW non-polarized classical signals at 1540.56 nm wavelength, with real-time coupling optimization achieved via a hill-climbing algorithm [84]. A schematic diagram of the experiment is shown in Fig. 9(a). Figure 9(b) displays the time evolution of averaged QBERs in both the X and Z bases, measured every 5 minutes to track polarization drift effects. The results demonstrate high QBER stability over continuous operation periods exceeding tens of hours at 37 km transmission distance. Therefore, it would be straightforward to restore the polarization to its initial configuration through intermittent recalibration procedures. In contrast, the long-distance loop (55 km) shows degraded stability with unpredictable QBER fluctuations, principally caused by its enhanced temperature sensitivity in environments. It is important to note that the increase of QBER is not significantly, almost on a minute scale. It would be straightforward to bring the polarization back to its initial value by active stabilization, which can be reliably achieved through automated compensation systems without reference laser [85]. Active stabilization is therefore indispensable for sustained stability, which can be reliably achieved through automated compensation systems [51, 52]. For example, the compensation module can be activated using QBER as a feedback signal [86], thereby refraining from “pollution” of the relatively strong reference light. To meet high-speed requirements, commercially available low-loss piezo-based polarization controller with response time in order of dozens of microseconds can be used to track polarization variations at rates approaching 10 rad/s [87]. Additionally, the nonlocality in photon−pair entanglement enables global polarization stabilization through local manipulation of just one photon.

6 Conclusion

In summary, we have performed a comprehensive analysis of polarization entanglement distribution contaminated with classical light in the same fiber link at telecom C-band. The properties of noise from classical light are characterized and subsequently employed to evaluate classical signal-induced QKD performance degradation. The simulation results indicate that the presence of classical light has a serious effect on maximum distance due to the increased uncorrelated noise. The optimal brightness of the quantum source decreases as the classical light power increases. Quantum/classical coexistence entanglement distribution achieves significantly enhanced performance in high-noise regimes, leveraging the inherent quantum correlations between entangled photons to achieve superior noise resilience. Through experimental validation of coexistence scenarios, we demonstrate that temporal correlation plays a key role in effective noise mitigation, highlighting the importance of optimizing the coincidence window for specific conditions. Meanwhile, the maximum key rate can be improved by using detectors with high detection efficiency, and by adjusting the source brightness and spectral filtering bandwidth within a certain range. Moreover, this quantum source can function as a heralded single-photon source — a key enabling technology for measurement-device-independent QKD systems [88, 89]. Due to its superior signal-to-noise ratio and compatibility with the classical-quantum co-transmission system in optical fibers, it represents a promising direction for future research.

Beyond this, quantum entanglement enables distinct pathways for further suppression of co-propagation noise. For instance, classical crosstalk in shared optical fibers can be mitigated by actively coordinating the relative orientations of entangled photons and coexisting classical signals, complemented by optimized local filtering that maintains maximal entanglement in the Bell-state portion while suppressing non-entangled terms [90, 91]. To address the case of highly unpolarized Raman scattering noise, the nonlocal dispersion cancellation effect can be employed to enhance the discrimination between true coincidence events and background noise [92]. In this situation, the uncorrelated noise undergoes significant temporal broadening exceeding the detector’s intrinsic timing resolution, enabling effective noise suppression through application of a suitable temporal window. These noise mitigation strategies offer novel approaches for implementing advanced quantum network functionalities beyond QKD, such as quantum conference key agreement [93, 94] and quantum secret sharing [95] in real-world scenarios.

7 Appendix A: Quantum bit error rate analysis

The imperfections of experimental system may cause error bits in entanglement-based QKD. Following Ref. [56 and 61], we assume that bit errors are mainly caused by background counts, intrinsic detector errors, and multiphoton events. The background counts include the random noise caused by the detector, stray light, and co-propagating classical light. The error probability of these events is e_0=\frac{1}{2}. The intrinsic detector error probability ({\mathrm{e}}^{pol}) quantifies the probability of non-ideal two-photon coincidences deviating from expected Bell-state correlations, arising from source imperfections, optical misalignment, and system instability. Multiphoton events include: (i) Alice and Bob detect different photon pairs, and (ii) double clicks. The error probability of either case will also be e_0=\frac{1}{2} under random bit assignment for double-click events. Let us consider the multiphoton-pair state

(A1)\begin{array}{c}\begin{array}{rl}& {\otimes }_{k=1}^N|{\mathrm{\Phi }}_{n_k}⟩={\otimes }_{k=1}^N\left(\frac{1}{\sqrt{n_k+1}}\right)\\ & \cdot \sum _{m_k=0}^{n_k}{|n_k-m_k⟩}_{H,A}{|m_k⟩}_{V,A}{|n_k-m_k⟩}_{V,B}{|m_k⟩}_{H,B}.\end{array}\end{array}

Using Eqs. (5) and (6), the overall gain for this state is given by

(A2)\begin{array}{c}\begin{array}{rl}{Q}_{\xi }=& \sum _{n_1,\cdots ,n_k,\cdots ,n_N}Y\left(n_1,\cdots ,n_k,\cdots ,n_N,t_{cc}\right)\prod _{k=1}^N{P}_k\left(n_k\right)\\ =& {\eta }_{t_{cc}}[1-\prod _{k=1}^N\frac{1}{{(1+{\eta }_B{\displaystyle \frac{{\xi }_k}{2}})}^2}-\prod _{k=1}^N\frac{1}{{(1+{\eta }_A{\displaystyle \frac{{\xi }_k}{2}})}^2}\\ & +\prod _{k=1}^N\frac{1}{{(1+{\eta }_A{\displaystyle \frac{{\xi }_k}{2}}+{\eta }_B{\displaystyle \frac{{\xi }_k}{2}}-{\eta }_A{\eta }_B{\displaystyle \frac{{\xi }_k}{2}})}^2}]\\ & +(D_A+N_A)\cdot [\prod _{k=1}^N\frac{1}{{(1+{\eta }_A{\displaystyle \frac{{\xi }_k}{2}})}^2}\\ & -\prod _{k=1}^N\frac{1}{{(1+{\eta }_A{\displaystyle \frac{{\xi }_k}{2}}+{\eta }_B{\displaystyle \frac{{\xi }_k}{2}}-{\eta }_A{\eta }_B{\displaystyle \frac{{\xi }_k}{2}})}^2}]\\ & +(D_B+N_B)\cdot [\prod _{k=1}^N\frac{1}{{(1+{\eta }_B{\displaystyle \frac{{\xi }_k}{2}})}^2}\\ & -\prod _{k=1}^N\frac{1}{{(1+{\eta }_A{\displaystyle \frac{{\xi }_k}{2}}+{\eta }_B{\displaystyle \frac{{\xi }_k}{2}}-{\eta }_A{\eta }_B{\displaystyle \frac{{\xi }_k}{2}})}^2}]\\ & +\prod _{k=1}^N\frac{(D_A+N_A)(D_B+N_B)}{{(1+{\eta }_A{\displaystyle \frac{{\xi }_k}{2}}+{\eta }_B{\displaystyle \frac{{\xi }_k}{2}}-{\eta }_A{\eta }_B{\displaystyle \frac{{\xi }_k}{2}})}^2}.\end{array}\end{array}

Just as in Section, when N\to \mathrm{\infty }, the singular value {\lambda }_k will approximate 1/N, it means that the average number of photons is identical across all modes, so we use \xi to denote the average number of photons when the number of modes approaches infinity, with \xi \simeq N\cdot {\xi }_k\simeq \frac{C^2}{{\hslash }^2}. Thus, we will obtain the following three sets of equations:

(A3)\begin{array}{c}\begin{array}{rl}& \prod _{k=1}^N\frac{1}{{(1+{\eta }_A{\displaystyle \frac{{\xi }_k}{2}})}^2}\approx {\mathrm{e}}^{-{\eta }_A\xi },\\ & \prod _{k=1}^N\frac{1}{{(1+{\eta }_B{\displaystyle \frac{{\xi }_k}{2}})}^2}\approx {\mathrm{e}}^{-{\eta }_B\xi },\\ & \prod _{k=1}^N\frac{1}{{(1+{\eta }_A{\displaystyle \frac{{\xi }_k}{2}}+{\eta }_B{\displaystyle \frac{{\xi }_k}{2}}-{\eta }_A{\eta }_B{\displaystyle \frac{{\xi }_k}{2}})}^2}\approx {\mathrm{e}}^{-{\eta }_A\xi -{\eta }_B\xi +{\eta }_A\xi {\eta }_B\xi }.\end{array}\end{array}

Substituting Eq. (A3) into Eq. (A2), we can easily obtain Eq. (8).

In the following calculation, we restrict our analysis to the case where both Alice and Bob project the state onto the Z basis, as the results for the X basis follow analogously by symmetry. Firstly, considering that the polarizations of the states detected by Alice and Bob are uniquely opposite, we refer to this detection event as a true single-click event [56]. This event will end up with an error probability of {\mathrm{e}}^{pol}, with an occurrence probability of

(A4)\begin{array}{c}\begin{array}{rl}\alpha =& \prod _{k=1}^N\left(\frac{1}{n_k+1}\right)\sum _{m_1...m_k...m_N}^{n_1...n_k...n_N}\\ & \{\left[1-{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\\ & \times \left[1-{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_B\right)}^{\sum _k{m}_k}\\ & +\left[1-{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\\ & \times \left[1-{\left(1-{\eta }_B\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\}.\end{array}\end{array}

Secondly, considering that the polarizations of the states detected by Alice and Bob are the same, we refer to this detection event as an error single-click event [56]. This event will yield an error probability of 1-{\mathrm{e}}^{pol}, and the probability of this event is

(A5)\begin{array}{c}\begin{array}{rl}\beta =& \prod _{k=1}^N\left(\frac{1}{n_k+1}\right)\sum _{m_1...m_k...m_N}^{n_1...n_k...n_N}\\ & \{\left[1-{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\\ & \times \left[1-{\left(1-{\eta }_B\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\\ & +\left[1-{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\\ & \times \left[1-{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_B\right)}^{\sum _k{m}_k}\}.\end{array}\end{array}

For all other cases, the error rate is e_0. Consequently, the error rate for the multi-photon-pair state is given by

(A6)\begin{array}{c}\begin{array}{rl}eY=& e_0\left(Y-\alpha -\beta \right)+{\mathrm{e}}^{pol}\alpha +\left(1-{\mathrm{e}}^{pol}\right)\beta \\ & +{\displaystyle \frac{1-2{\mathrm{e}}^{pol}}{4}}\left(\alpha -\beta \right)\left[1-\left(1-N_{A,tol}\right)\left(1-N_{B,tol}\right)\right],\end{array}\end{array}

where Y is shown in Eq. (6), and N_{A(B),tol}=D_{A(B)}+N_{A(B)} is the total noise count rate in Alice’s (Bob’s) detection system, consisting of the dark count rate D_{A(B)} and co-propagation noise rate N_{A(B)}. The error rate for the state with a definite number of photons, n_1,\dots ,n_k,\dots ,n_N, can be written as

\begin{array}{rl}eY=& e_{0}Y-\left(e_0-{\mathrm{e}}^{pol}-\frac{1-2{\mathrm{e}}^{pol}}{4}\mathrm{\Delta }N\right)\\ & \times \{{\prod }_{k=1}^N\left(\frac{1}{n_k+1}\right)\end{array}

(A7)\begin{array}{c}\begin{array}{rl}& \sum _{m_1...m_k...m_N}^{n_1...n_k...n_N}\{\left[1-{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\\ & \times \left[1-{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_B\right)}^{\sum _k{m}_k}\\ & +\left[1-{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\\ & \times \left[1-{\left(1-{\eta }_B\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\}\\ & -{\prod }_{k=1}^N\left(\frac{1}{n_k+1}\right)\\ & \sum _{m_1...m_k...m_N}^{n_1...n_k...n_N}\{\left[1-{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\\ & \times \left[1-{\left(1-{\eta }_B\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\\ & +\left[1-{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\\ & \times \left[1-{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_B\right)}^{\sum _k{m}_k}\}\},\end{array}\end{array}

where \mathrm{\Delta }N=N_{A,tol}+N_{B,tol}-N_{A,tol}{N}_{B,tol}. By summing up the photon numbers in Eq. (A7) and accounting for both the photon number distribution function in Eq. (5) and the overall gain in Eq. (8), we can obtain the overall QBER

(A8)\begin{array}{c}\begin{array}{rl}{E}_{\xi }=& \frac{{\displaystyle \sum _{n_1,\cdots ,n_k,\cdots ,n_N}^{\mathrm{\infty }}eY\left(n_1,\cdots ,n_k,\cdots ,n_N\right){\displaystyle \prod _{k=1}^N{P}_k(n_k)}}}{Q_{\xi }}\\ =& e_0-{\eta }_{t_{cc}}\left[{\mathrm{e}}^{-\frac{\xi }{2}\left({\eta }_A+{\eta }_B-{\eta }_A{\eta }_B\right)}-{\mathrm{e}}^{-\frac{\xi }{2}\left({\eta }_A+{\eta }_B\right)}\right]\\ & \times \frac{2\left(e_0-{\mathrm{e}}^{pol}-{\displaystyle \frac{1-2{\mathrm{e}}^{pol}}{4}}\mathrm{\Delta }N\right)}{Q_{\xi }}.\end{array}\end{array}

8 Appendix B: Influence of polarization variations on QBER

As mentioned earlier, photon polarization state evolution exhibits strong dependence on environmental conditions and optical channel characteristics. To comprehensively evaluate the QBER variations induced by polarization drifts, we model the probability of the intrinsic detector error as {\mathrm{e}}_2^{pol}={\mathrm{e}}^{pol}+\mathrm{\Delta }{e}_p, where \mathrm{\Delta }{e}_p quantifies the additional error probability arising from polarization drift. In this case, the overall QBER can be expressed as

(B1)\begin{array}{c}\begin{array}{rl}{E}_{\xi }=& e_0-{\eta }_{t_{cc}}\left[{\mathrm{e}}^{-\frac{\xi }{2}\left({\eta }_A+{\eta }_B-{\eta }_A{\eta }_B\right)}-{\mathrm{e}}^{-\frac{\xi }{2}\left({\eta }_A+{\eta }_B\right)}\right]\\ & \times \frac{2\left(e_0-{\mathrm{e}}_2^{pol}-{\displaystyle \frac{1-2{\mathrm{e}}_2^{pol}}{4}}\mathrm{\Delta }N\right)}{Q_{\xi }}.\end{array}\end{array}

In optical fibers with negligible polarization-dependent loss, the polarization state transformation between input and output can be modeled as a unitary operation \hat{U}\in SU\left(2\right), preserving the photon’s purity while inducing general elliptical polarization transformations [86, 96]. Considering the rectilinear basis as a reference, photons emitted from the source experience random polarization drift before detection, where fiber-induced polarization changes can be also modeled by a Jones matrix [97]. To characterize the influence of polarization variations on the polarization entanglement distribution systems, we denote the Jones matrixes characterizing the optical fiber channel to Alice’s and Bob’s detection system as

(B2)\begin{array}{c}{J}_{Alice}=\left(\begin{array}{cc}{a}_1& b_1\\ -b_1^{\ast }& a_1^{\ast }\end{array}\right),J_{Bob}=\left(\begin{array}{cc}{a}_2& b_2\\ -b_2^{\ast }& a_2^{\ast }\end{array}\right),\end{array}

where a_{1(2)} and b_{1(2)} are plural, which describing the influence of optical fibers on the phase and polarization of photons, and they satisfy the following equation: {\left|a_{1(2)}\right|}^2+{\left|b_{1(2)}\right|}^2=1. After that, the multiphoton-pair state before detection can be modified as

(B3)\begin{array}{c}\begin{array}{rl}{\otimes }_{k=1}^N|{\mathrm{\Phi }}_{n_k}⟩=& {\otimes }_{k=1}^N[\frac{1}{\sqrt{n_k+1}}\sum _{m_k=0}^{n_k}\\ & \times (a_1{|n_k-m_k⟩}_{H,A}-b_1^{\ast }{|n_k-m_k⟩}_{V,A})\\ & \times (a_1^{\ast }{|m_k⟩}_{V,A}+b_1{|m_k⟩}_{H,A})\\ & \times (a_2^{\ast }{|n_k-m_k⟩}_{V,B}+b_2{|n_k-m_k⟩}_{H,B})\\ & \times (a_2{|m_k⟩}_{H,B}-b_2^{\ast }{|m_k⟩}_{V,B})].\end{array}\end{array}

For the state given in Eq. (B3), the error-free single-click probability is

(B4)\begin{array}{c}\begin{array}{rl}{\alpha }^{\prime }=& {\prod }_{k=1}^N\left(\frac{1}{n_k+1}\right)\sum _{m_1...m_k...m_N}^{n_1...n_k...n_N}\\ & \{({\left|a_1\right|}^4{\left|a_2\right|}^4-{\left|a_1\right|}^4{\left|a_2\right|}^2{\left|b_2\right|}^2+{\left|a_1\right|}^2{\left|a_2\right|}^4{\left|b_1\right|}^2+{\left|a_1\right|}^2{\left|b_1\right|}^2{\left|b_2\right|}^4-{\left|a_2\right|}^2{\left|b_1\right|}^4{\left|b_2\right|}^2+{\left|b_1\right|}^4{\left|b_2\right|}^4)\\ & \times \left[1-{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\left[1-{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_B\right)}^{\sum _k{m}_k}\\ & +({\left|a_1\right|}^4{\left|a_2\right|}^4+{\left|a_1\right|}^4{\left|a_2\right|}^2{\left|b_2\right|}^2-{\left|a_1\right|}^2{\left|a_2\right|}^4{\left|b_1\right|}^2-{\left|a_1\right|}^2{\left|b_1\right|}^2{\left|b_2\right|}^4+{\left|a_2\right|}^2{\left|b_1\right|}^4{\left|b_2\right|}^2+{\left|b_1\right|}^4{\left|b_2\right|}^4)\\ & \times \left[1-{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\left[1-{\left(1-{\eta }_B\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\\ & +(-{\left|a_1\right|}^4{\left|b_2\right|}^4-{\left|a_1\right|}^4{\left|a_2\right|}^2{\left|b_2\right|}^2-{\left|a_1\right|}^2{\left|a_2\right|}^4{\left|b_1\right|}^2-{\left|a_1\right|}^2{\left|b_1\right|}^2{\left|b_2\right|}^4-{\left|a_2\right|}^2{\left|b_1\right|}^4{\left|b_2\right|}^2-{\left|b_1\right|}^4{\left|a_2\right|}^4)\\ & \times \left[1-{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\left[1-{\left(1-{\eta }_B\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\\ & +(-{\left|a_1\right|}^4{\left|b_2\right|}^4+{\left|a_1\right|}^4{\left|a_2\right|}^2{\left|b_2\right|}^2+{\left|a_1\right|}^2{\left|a_2\right|}^4{\left|b_1\right|}^2+{\left|a_1\right|}^2{\left|b_1\right|}^2{\left|b_2\right|}^4+{\left|a_2\right|}^2{\left|b_1\right|}^4{\left|b_2\right|}^2-{\left|b_1\right|}^4{\left|a_2\right|}^4)\\ & \times \left[1-{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\left[1-{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_B\right)}^{\sum _k\left(m_k\right)}\}.\end{array}\end{array}

Similarly, the probability of getting an erroneous single-click event can be expressed as

\begin{array}{rl}& \\ {\beta }^{\prime }=& {\prod }_{k=1}^N\left(\frac{1}{n_k+1}\right)\sum _{m_1...m_k...m_N}^{n_1...n_k...n_N}\\ & \{(-{\left|a_1\right|}^4{\left|b_2\right|}^4+{\left|a_1\right|}^4{\left|a_2\right|}^2{\left|b_2\right|}^2-{\left|a_1\right|}^2{\left|a_2\right|}^4{\left|b_1\right|}^2-{\left|a_1\right|}^2{\left|b_1\right|}^2{\left|b_2\right|}^4+{\left|a_2\right|}^2{\left|b_1\right|}^4{\left|b_2\right|}^2-{\left|b_1\right|}^4{\left|a_2\right|}^4)\\ & \times \left[1-{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\left[1-{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_B\right)}^{\sum _k{m}_k}\\ & +(-{\left|a_1\right|}^4{\left|b_2\right|}^4-{\left|a_1\right|}^4{\left|a_2\right|}^2{\left|b_2\right|}^2+{\left|a_1\right|}^2{\left|a_2\right|}^4{\left|b_1\right|}^2+{\left|a_1\right|}^2{\left|b_1\right|}^2{\left|b_2\right|}^4-{\left|a_2\right|}^2{\left|b_1\right|}^4{\left|b_2\right|}^2-{\left|b_1\right|}^4{\left|a_2\right|}^4)\\ & \times \left[1-{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\left[1-{\left(1-{\eta }_B\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\end{array}

(B5)\begin{array}{c}\begin{array}{rl}& +({\left|a_1\right|}^4{\left|a_2\right|}^4+{\left|a_1\right|}^4{\left|a_2\right|}^2{\left|b_2\right|}^2+{\left|a_1\right|}^2{\left|a_2\right|}^4{\left|b_1\right|}^2+{\left|a_1\right|}^2{\left|b_1\right|}^2{\left|b_2\right|}^4+{\left|a_2\right|}^2{\left|b_1\right|}^4{\left|b_2\right|}^2+{\left|b_1\right|}^4{\left|b_2\right|}^4)\\ & \times \left[1-{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\left[1-{\left(1-{\eta }_B\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\\ & +({\left|a_1\right|}^4{\left|a_2\right|}^4-{\left|a_1\right|}^4{\left|a_2\right|}^2{\left|b_2\right|}^2-{\left|a_1\right|}^2{\left|a_2\right|}^4{\left|b_1\right|}^2-{\left|a_1\right|}^2{\left|b_1\right|}^2{\left|b_2\right|}^4-{\left|a_2\right|}^2{\left|b_1\right|}^4{\left|b_2\right|}^2+{\left|b_1\right|}^4{\left|b_2\right|}^4)\\ & \times \left[1-{\left(1-{\eta }_A\right)}^{\sum _k{m}_k}\right]{\left(1-{\eta }_A\right)}^{\sum _k\left(n_k-m_k\right)}\left[1-{\left(1-{\eta }_B\right)}^{\sum _k\left(n_k-m_k\right)}\right]{\left(1-{\eta }_B\right)}^{\sum _k\left(m_k\right)}\}.\end{array}\end{array}

After some algebra, the over QBER accounting for polarization drift effects can be expressed as

(B6)\begin{array}{c}\begin{array}{rl}{E}_{{\xi }_{dr}}^{\prime }{Q}_{\xi }=& e_0{Q}_{\xi }-2\left(e_0-{\mathrm{e}}^{pol}-\frac{1-2{\mathrm{e}}^{pol}}{4}\mathrm{\Delta }N\right)\\ & \times \left[{\left|a_1\right|}^4+{\left(1-{\left|a_1\right|}^2\right)}^2\right]\left[{\left|a_2\right|}^4+{\left(1-{\left|a_2\right|}^2\right)}^2\right]\\ & \times {\eta }_{t_{cc}}\left[{\mathrm{e}}^{-\frac{\xi }{2}\left({\eta }_A+{\eta }_B-{\eta }_A{\eta }_B\right)}-{\mathrm{e}}^{-\frac{\xi }{2}\left({\eta }_A+{\eta }_B\right)}\right].\end{array}\end{array}

Then we can obtain

(B7)\begin{array}{c}\begin{array}{rl}{E}_{{\xi }_{dr}}^{\prime }=& e_0-\frac{2\left(e_0-{\mathrm{e}}^{pol}-\frac{1-2{\mathrm{e}}^{pol}}{4}\mathrm{\Delta }N\right)}{Q_{\xi }}\\ & \times \left[{\left|a_1\right|}^4+{\left(1-{\left|a_1\right|}^2\right)}^2\right]\left[{\left|a_2\right|}^4+{\left(1-{\left|a_2\right|}^2\right)}^2\right]\\ & \times {\eta }_{t_{cc}}\left[{\mathrm{e}}^{-\frac{\xi }{2}\left({\eta }_A+{\eta }_B-{\eta }_A{\eta }_B\right)}-{\mathrm{e}}^{-\frac{\xi }{2}\left({\eta }_A+{\eta }_B\right)}\right].\end{array}\end{array}

Eqs. (B1) and (B7) represent two formulations for the overall QBER under the influence of polarization variations. Through comparison, we can relate the polarization-dependent error probability \mathrm{\Delta }{e}_p to the Jones matrix elements as

(B8)\begin{array}{c}\begin{array}{rl}\mathrm{\Delta }{e}_p=& \frac{2\left(e_0-{\mathrm{e}}^{pol}-{\displaystyle \frac{1-2{\mathrm{e}}^{pol}}{4}}\mathrm{\Delta }N\right)}{\mathrm{\Delta }N-2}\\ & \times \{\left[{\left|a_1\right|}^4+{\left(1-{\left|a_1\right|}^2\right)}^2\right]\left[{\left|a_2\right|}^4+{\left(1-{\left|a_2\right|}^2\right)}^2\right]\\ & -1\}.\end{array}\end{array}

Since the noise count rate N_{A,tol} and N_{B,tol} are much less than 1, Eq. (B8) can be simplified to

(B9)\begin{array}{c}\begin{array}{rl}\mathrm{\Delta }{e}_p=& \left(e_0-{\mathrm{e}}^{pol}\right)\\ & \times \{1-\left[{\left|a_1\right|}^4+{\left(1-{\left|a_1\right|}^2\right)}^2\right][{\left|a_2\right|}^4\\ & +{\left(1-{\left|a_2\right|}^2\right)}^2]\}.\end{array}\end{array}

According to Eq. (B9), one can discuss how the Jones matrix elements influence the error probability \mathrm{\Delta }{e}_p. For example, if the Jones matrix is an identity matrix, i.e., |a_1|=|a_2|=1,|b_1|=|b_2|=0, that means there is no polarization drift in the fiber, thus \mathrm{\Delta }{e}_p=0. If |a_1|=|a_2|=0,|b_1|=|b_2|=1, it corresponds to complete polarization reversal for both Alice and Bob, which similarly results in \mathrm{\Delta }{e}_p=0. If |a_1{|}^2=|a_2{|}^2=0.5, |b_1{|}^2=|b_2{|}^2=0.5, the \mathrm{\Delta }{e}_p will reach its maximum value \frac{3}{4}(e_0-{\mathrm{e}}^{pol}). It is obvious that the diversity between the Jones matrix and the identity matrix is closely related to the polarization basis misalignment between Alice and Bob. The distance between them can be selected as a feedback signal to implement a continuous polarization-basis tracking scheme [86].

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