The development of quantum network (QN) is crucial for exploring applications beyond its classical counterpart, such as distributed and blind computing^1,2, enhanced sensing^3,4, and ultra-secure communications^5,6. To date, QNs of various topologies have been developed with trusted relays or active/passive routing^7-17. Among them, fully and simultaneously connected QN architecture is desirable due to its practicality and scalability^12-16. In this architecture, because of the nonlocal properties of quantum states distributed in networks, dedicated entanglement between any pair of users can be established even without a direct optical link¹⁷. Currently, bringing QNs closer to practice is essential for scaling the connection of quantum nodes. A promising approach for the implementation of scalable QNs is to leverage off-the-shelf fiber infrastructures and technologies, and transmit both quantum and classical signals in shared fibers, as this would efficiently reduce the costs.

Up till now, a series of quantum-classical coexistence investigations have been implemented in fiber-based quantum communications^18-26. However, most of these studies have focused solely on the point-to-point quantum key distribution (QKD) protocol, and the coexistence of multiuser entanglement distribution and classical light has not yet been fully exploited. The main challenge for hybrid quantum-classical communication is the strong noise induced by high-power classical light. Similar to classical communication, interaction with noise will eventually reduce the channel capacity, becoming a limiting factor in terms of both rate and distance in quantum communication. It is therefore crucial to design advanced protocols that can withstand large amounts of noise²⁷.

High-dimensional (HD) entanglement is often considered to have the potential to provide stronger violations of Bell inequalities and better noise tolerance in quantum communications^28,29. Despite all this, the high noise resistance of HD entanglement has been recently demonstrated in realistic scenarios³⁰. This is primarily because the HD quantum states are difficult to control and measure, and the increase of system dimensions often introduces additional noise^31,32. To implement entanglement-based HD-QKD more effectively, a simultaneous multiple subspace coding protocol is proposed, enabling a secret key even in extremely noisy conditions³³. Then the theoretical prediction has been verified in a proof-of-principle implementation using path entanglement with multi-outcome measurements³⁴. Nonetheless, the scalability of this approach is limited due to the increase of detector number and the difficulty in achieving phase stability. Fortunately, the subspace noise-resistant protocol has been realized in ref.³⁵ using polarization-frequency hyperentanglement, which is practically deployable and naturally suitable for fiber QNs. This pioneer scheme offers three advantages. Firstly, the d × d-dimensional hyperentangled system can be divided into l subspaces of size 2 × 2 (polarization entanglement), where d = l × 2. The polarization qubit subspace can be used for coding and achieves the highest noise resistance³³, whereas the frequency qudit subspace is used to dilute dimension-dependent noise. Secondly, polarization is easy to control and analyze without nested interferometers and has mature automatic polarization feedback system. Additionally, polarization encoding has been operated continuously in installed fiber networks for several hours without active compensation¹³. Thirdly, the multiplexing of frequency degree of freedom, which is compatible with off-the-shelf fiber network infrastructures, can not only improve data rates in classical communications³⁶ but also provide wide connectivity for QNs.

Here, we leverage and expand upon these advantages to establish a fully connected three-user QN that operates in tandem with classical light, utilizing a high-efficiency semiconductor chip. First of all, the entangled state is generated via modal phase matching (PM) within the Bragg reflection waveguide (BRW) structure, with the adoption of AlGaAs-/GaAs-based materials. The quantum source achieves PM by utilizing bounded total internal reflection (TIR) modes formed between high- and low-index claddings, along with quasi-bounded BRW modes guided though transverse Bragg reflections at the interface between core and period claddings^37,38. Meanwhile, the source is chip-based, providing stable and alignment-free operations³⁹. It is complementary metal oxide semiconductor-compatible and is well developed for various devices such as lasers and modulators^38,40-45. The results show that the monolithic source maintains the polarization entanglement fidelity of over 96% across a 42-nm bandwidth, with a brightness of 1.2 × 10⁷ Hz mW⁻¹. Furthermore, the polarization-entangled photons are partitioned and are used for the characterization of its increased resilience to noise. In the experiments, the noise is introduced by coexisting a classical light with quantum signals in the same fiber, which is extremely practical for installed fiber networks. Subsequently, a fully connected three-user quantum communication is achieved in a noisy network by using three pairs of polarization-entangled photons. Each user analyzes and decodes the polarization qubits using two unbalanced polarization-maintaining interferometers (UPMIs) and a single detector via time multiplexing. Finally, the generated secure keys are used for image encryption and quantum secret sharing (QSS). Overall, these results, combined with the utilization of subspace coding technology, present a promising pathway for advanced QN applications in deployed fibers.

Firstly, the protocol developed are briefly reviewed in ref.³⁵. Assuming entanglement in both polarization and frequency degrees, the ideal entangled state can be expressed as⁴⁶:

where $ \left|\Psi^{+}\right\rangle=\frac{1}{\sqrt{2}}(|\mathrm{HV}\rangle+|\mathrm{VH}\rangle)$ and $ |\psi\rangle=\sum_{\mathrm{n}=1}^{\mathrm{N}} \frac{1}{\sqrt{N}}\left|\omega_{\mathrm{s}, \mathrm{n}}\right\rangle\left|\omega_{\mathrm{i}, \mathrm{N}-\mathrm{n}}\right\rangle$, with $ \omega_{\mathrm{s}, \mathrm{n}}=\omega_{0}+n \Delta \omega$ and $ \omega_{\mathrm{i}, \mathrm{N}-\mathrm{n}}=\omega_{\mathrm{N}}-n \Delta \omega$ represent signal and idler photons chosen from the spectrum, respectively, such that the sum of $ \omega_{0}$ and $\omega_{\mathrm{N}}$ is equal to the pump frequency. N is the total number of frequency-bin pairs selected. $ |\mathrm{H}\rangle /|\mathrm{V}\rangle$ represents the horizontally or vertically polarized states. As shown in Fig. 1a, $ \rho_{1}$ transmits over-noisy channels that can be thought of as pairs of frequency-correlated photons passing through independent links and are measured separately. Some of the most usual noise for qubits are discussed now: amplitude-damping and white noise. The influence of white noise on the state can be described as follows:

where $p$ is the noise portion and $ I_{2}\left(I_{\mathrm{N}}\right)$ represents the completely mixed state in the polarization (frequency) subspace. After tracing over the frequency, the state can be written as an isotropic state

As can be seen, the white noise leads to accidental coincidences in experiments and can be equivalently divided by the frequency subspace dimension N. This is the physical origin of noise resistance in this protocol. By choosing appropriate frequency channels, the probability of measuring false coincidences between noise-noise photons or noise-signal photons is reduced, while the probability of measuring a true coincidence between correlated photons remains essentially unchanged. Therefore, the additional quantum correlation in frequency can be used to distill the signal photons from noise photons. Furthermore, physical noise for quantum systems with one quantum state is generally characterized by the Kraus representation⁴⁷. The amplitude-damping noise channel represents the dissipative interaction between the qubit and its environment. The corresponding Kraus operators are as follows:

where, $\gamma_{i}$ (i = 1, 2) denotes the noise portion. Therefore, the noise on $ \rho_{2}^{\prime}$ can be modeled by

The degree of entanglement of two-qubit states can be described by various entanglement measures including: the relative entropy of entanglement, the Peres-Horodecki negativity, and the Wootters concurrence. A comprehensive quantitative analysis of the relationship between the three entanglement measures can be found in ref.⁴⁸. Among them, the entanglement negativity⁴⁹, which is a measure of entanglement cost under operations preserving the positivity of partial transpose, is a well-known tool for quantifying bipartite quantum correlations⁵⁰. Given the aforementioned density matrix of a composite quantum system, the entanglement negativity^51,52 between two bipartitions of the system is given as follows:

where

For simplicity, in Fig. 1b, we model the effects of amplitude-damping noise with coefficients $ \gamma_{1}=\gamma_{2}=\gamma \in[0.1,0.3]$, and white noise with portion p on such state. It is clear that the effect of noise can be significantly reduced by increasing N, which indicates quantum correlation is an effective resource to resist noise. It is worth mentioning that the simple noise models are analyzed here only serve as a quantitative example to explain why the increased noise resistance can be expected by applying hyperentanglement. A more quantitative analysis of the precise role of noise in photonic entanglement has been performed in ref.⁵³. Nonetheless, we do not assume any noise model when quantitatively analyzing the following realistic quantum–classical signals’ coexistence in fiber network.

Fig. 2a shows the schematic of experimental setup, containing three parts: photon-pair source, classical–quantum link and wavelength-selective switch (WSS), and polarization analysis module. In the source part, a single-mode fiber-coupled continuous-wave (CW) laser centered at 780.08-nm pumps the semiconductor BRW source. 1% power of the laser is split to the optical spectrum analyzer for wavelength calibration and a fiber polarization controller is used to adjust the pump polarization before injection into the source. The source consists of a core layer sandwiched by six periods upper and lower Bragg layers and has a length of 4 mm. Broadband polarization–entangled photon pairs can be directly generated by degenerate type-II spontaneous parametric down conversion (SPDC) due to the low material birefringence (see Supplementary Materials 1 for detailed description). The lack of emitted time information of photons in the SPDC process leads to energy–time entanglement⁵⁴. Light is coupled into and out from the chip by lens-tapered fibers mounted on high-precision servo motors. The total insertion loss of the chip is about 11 dB, including both input–output coupling loss and propagation loss in the chip for the TIR modes. To stabilize the pump transmission, we use the rejected pump light filtered by a 980/1550 dense wavelength division multiplexer (DWDM) as a feedback signal for the motor and adopt hill-climbing algorithm to optimize the coupling under unstable laboratory conditions. Due to energy conservation during the SPDC process, the polarization-entangled photons exhibit anti-correlation in frequency. The correlated photons and an external noise source are then multiplexed into the same fiber link via DWDM. In the experiment, noise is introduced by co-transmitting a 1591.26-nm classical laser with quantum signals through a 3-km-long fiber, representing a realistic scenario for a quantum communication system operating on public DWDM networks. The coexistence of classical and quantum signals in the same fiber can lead to some problems that may affect the performance of quantum communication. The dominant sources of noise are crosstalk from the classical light when the used DWDM has insufficient isolation and Raman scattering, which occurs due to the inelastic photon-phonon interaction^55,56. In this experiment, the wavelength of classical light is far away from that of quantum signals; therefore, the crosstalk of the non-adjacent channels can be eliminated by the use of appropriate filters on the quantum receivers. However, the broadband Raman noise cannot be spectrally filtered out as it is in-band with the quantum signals. In order to obtain the Raman cross section for different signal and idler channels, the measurement results are obtained using a laser centered at 1591.26 nm in a single-mode fiber (see Supplementary Materials 3). By using a WSS, six wavelength grids of the signal (ranging from 1554.1 nm to 1557.1 nm) and idler (ranging from 1563.4 nm to 1566.4 nm) spectra were postselected. Each selected grid has a full-width half maximum bandwidth of 0.4 nm and wavelength interval of 0.6 nm. The polarization analysis module, along with each channel, comprises a quarter-wave plate, a half-wave plate, a UPMI, and a SPD. In the interferometer, two orthogonal polarizations are connected to polarization-maintaining fibers of different lengths, converting them into arrival time information for the photons. The number of coincidence counts in four configurations are consequently obtained in one data accumulation. The single-photon detection events are recorded using a field-programmable-gate-array-based timetag unit. From these timetag records, both single counts and coincidence counts are extracted. The d-dimensional Hilbert space is split into d/2 mutually exclusive polarization subspaces. Fig. 2b shows the 6 × 6 joint spectrum intensity for four different classical laser powers in the Z = {H, V} and X = {A, D} bases with an integration time of 120 s. An input pump power of 30 mW and a coincidence window of 0.5 ns were set. As expected, increasing the power of the classical laser results in a higher number of noise photons, which randomly spread the whole selected spectral range, while the correlated signals distributed in the diagonal elements remain almost constant. From the project measurements on the two mutually unbiased bases, a lower bound on the fidelity to the Bell state can be estimated⁴⁴. Fig. 2b shows the averaged lower-bound fidelity of the six frequency subspaces shown diagonally. The decrease in the fidelity can provide an intuitive explanation for the state becoming more mixed in more noisy channels (see Supplementary Materials 3 for a detailed theoretical analysis). Fig. 2c shows the averaged quantum bit error rates (QBERs) in Z and X bases of various division numbers for different classical powers. The QBER is determined by using the raw coincidence counts in each basis: QBER_z = (C_HH+C_VV)/(C_HH+ C_HV+ C_VH+ C_VV) and QBER_x = (C_DA+ C_AD)/(C_DA+ C_DD+ C_AD+C_AA). To study the effects of coarse-grained frequency division, the 6 × 6 matrix is summed to reconstruct 1 × 1, 2 × 2, and 3 × 3 matrices for the estimation of QBERs. The results show that as the number of mutually exclusive subspaces increases, QBERs decrease dramatically, especially over highly noisy quantum channels, indicating stronger noise resilience for larger N in QNs. Fig. 2d shows the lower-bound concurrence calculated by $C \geq V_{\mathrm{D} / \mathrm{A}}+V_{\mathrm{H} / \mathrm{V}}-1$ ⁵⁷ as a function of division number with four classical light powers. All the experimental results are in good agreement with theoretical predictions (colored solid lines) calculated through the analysis in Supplementary Materials 3. The increased degree of entanglement for large N implies nonlocality distillation, which also has the potential application to improve the noise resistance in device-independent QKD⁵⁸.

As a second series of measurements for a full use of the broad entanglement bandwidth and a complete demonstration of noise resilience in a quantum communication network, entanglement distribution for a fully connected three-user network is performed. As shown in Fig. 3, the central server hosts a broadband polarization-entangled photon pairs and selects three channel pairs {CH32, CH13}, {CH11, CH09}, and {CH34, CH30} via multiplexing and demultiplexing. Each user is assigned a combination of wavelengths as indicated in the detection module. On the network, Alice is connected via a 4.075-km spool with a loss of 1.9 dB, Bob and Charlie use 4.072-km (1.8 dB) and 3.054-km (1.2 dB) spools, respectively. Again, the introduction of noise into the fiber channel is conducted with the adoption of a classic laser. The launch power of the classical laser is set as −23 dBm to simulate the two particular modules (Finisar FWLF-1631-xx), where each module has a receiver sensitivity of −28 dBm to guarantee a bit error rate of <10^{-12 55}. At the receiving stage, photons incident on the polarization analysis module (up: H/V, down: A/D) are passively split by a 50 : 50 BS for basis choice. For each basis, the UPMI is adopted to identify polarization by the relative arrival times of photons. An additional delay is added between the two bases to further distinguish the basis information (see Supplementary Materials 4 for detailed description). The two outputs by a 50 : 50 coupler combined to reduce the detector number at an expense of a 3-dB loss. The channel is distinguished by implementing wavelength-dependent time multiplexing as the method used in ref.¹⁵. Alternatively, one can combine the two single-mode fibers into one multi-mode output⁵⁹. Due to the lack of an additional SPD, the two-user link is run separately, while the relative delay is added to make out different combinations of users. The time-correlation histograms corresponding to the highest recorded QBER for the three links are displayed in Fig. 4a. The simultaneous projection measurements on the Z and X bases are indicated by the marked 16 coincidence combinations. Fig. 4b shows the evolutions of the averaged QBERs and sifted key rates per minute with 100-GHz DWDMs in more than 65 h from a laboratory test. Steep spikes in the figures are mainly caused by room-temperature variations, which lead to perturbations in the coupling efficiency and the polarization states transmitted in the fibers. For comparison, we also recorded data for half an hour using 200-GHz DWDMs. The averaged QBERs are 11.07%, 13.51%, and 12.67% (denoted by black dashed lines), which are all above the 11% QBER threshold and are much larger than those of 100 GHz DWDMs {5.63%, 6.58%, 7.21%}. These results clearly demonstrate the practical feasibility of this quantum correlation assisted multi-user entanglement distribution in real-world telecommunication fiber networks.

To maximize the finite secret key length, we follow the security analysis and use the formula defined in ref.⁶⁰. For standard QKD security definition, if a protocol is $ \varepsilon_{1}$ correct and $ \varepsilon_{2}$ secret, it can be defined that the protocol is $ \varepsilon_{Q K D}$ secure with $ 0<\varepsilon_{1}+\varepsilon_{2} \leq \varepsilon_{\mathrm{QKD}}$, satisfying the following equation:

Assuming that the finite key length of implementation is $ l=\lfloor\alpha m\rfloor $, where m is the sifted key length and 0 < α < 1. The error functions due to parameter estimation (pe) and privacy amplification (pa) can be defined as follows:

and

where the correctness error is $ 2^{-t}=10^{-(s+2)} $ and security parameter is $ \varepsilon_{\mathrm{QKD}}=10^{-s} $; f is the error correction efficiency shown in Supplementary Materials 5. Therefore, we numerically maximize $ l$ by optimizing { $ \alpha, \beta, \nu, \xi$} while conforming to the inequality Eq. (8) with fixed block length and security level s = 9. It is numerically finds the parameter values {0.262, 0.080, 0.015, 0.013} for Alice and Bob, {0.149, 0.100, 0.014, 0.012} for Bob and Charlie, and {0.191, 0.09, 0.015, 0.013} for Alice and Charlie. The allowed longest keys are 388.56 kb, 199.72 kb, and 254.63 kb.

Utilization of the secure keys generated facilitated the encryption, transmission, and decryption of images between each pair of users. Diverging from the commonly adopted approach of directly encrypting images using XOR operations, the current experiment incorporates the Shannon confusion-diffusion architecture to safeguard the encrypted images against image-specific attacks, such as cropping attacks^61,62. In this architecture, confusion involves shuffling the pixel positions without altering their values, whereas diffusion entails sequentially modifying the pixel values using secret keys (see Supplementary Materials 6 for more details). In this experiment, as illustrated in Fig. 5, each sender performs three rounds of confusion operations, succeeded by two rounds of diffusion and confusion operations to encrypt the original image. Subsequently, the encrypted image is transmitted to the receiver via the classical channel. The receiver performs the inverse transformation of encryption process to obtain the decrypted image.

Unlike numerous QSS protocols that rely on the sequential transmission of quantum states between players, with each player encoding their data into the states using quantum transformations^63-67, the proposed strategy is capable of generating a set of unconditionally secure keys between any two users. This enables the direct establishment of a QSS network that empowers a dealer to share secret messages with selected players, consequently enhancing the practical feasibility and commercial potential of QSS. Fig. 6 illustrates the implementation of a three-user QSS network. In this network, the dealer, Alice, encrypts a secret message M by performing $ \mathrm{M} \oplus \mathrm{Key}_{\mathrm{AB}} \oplus \mathrm{Key}_{\mathrm{AC}}$, and transmits the resulting encrypted message C to two players, Bob and Charlie, via the classical channels. By sharing their secret keys and decrypting the received message using the equation $\mathrm{C} \oplus \mathrm{Key}_{\mathrm{AB}} \oplus \mathrm{Key}_{\mathrm{AC}}$, the players are able to reconstruct the secret message, while any attempt to obtain useful information without cooperation between the players will be futile.

In summary, a non-ideal quarter-wavelength BRW with high figure-of-merit modal overlap between interacting fields for photon-pair generation was designed and fabricated in this work. The broadband polarization entanglement with high fidelity in the telecom band can be directly generated due to the low material birefringence. The potential of subspace coding for mitigating noise in QNs was demonstrated with this source. The noise resilience has been analyzed theoretically and experimentally, showing that the QBER decreases significantly as the number of frequency bins N increases, particularly in highly noisy channels. This implies that quantum correlation can effectively overcome physical noise and increase the channel capacity, even under the extreme noise conditions that would otherwise preclude quantum communication. In addition, a fully connected three-user noisy network has been demonstrated by multiplexing 3 pairings of DWDM channels so that polarization entanglement can be shared between each possible two-party link. The broad spectrum, combined with considerable brightness, makes the BRW-entangled source highly promising for large-scale, wavelength-multiplexed, fully connected QNs. Additionally, by applying time-multiplexing in the polarization analysis module, the number of SPDs required has been reduced to one per user on the network. Finally, image encryption and QSS with the secure keys generated among each user have been performed by taking into account of the finite key effect. With the advancements of fabrication technology, it is possible to integrate laser-demultiplexing/-multiplexing and wavelength-demultiplexing/-multiplexing modules onto a single chip, thereby reducing the complexity and connection losses associated with off-chip individual optical components. This work lays the foundation for a turn-key solution for large-scale QNs that are compatible with fiber optical communication.

In experiments, the highly versatile modal birefringence in BRW enabled the PM of the three modalities^68,69. Here the properties of biphotons were considered to generate from a 4-mm non-ideal quarter-wavelength BRW with high modal overlap in the process of degenerate type-II SPDC⁷⁰, i.e., one TE-polarized pump photon at frequency $\omega_{\mathrm{p}}$ is converted into a pair of cross-polarized signal and idler photons at frequencies $\omega_{\mathrm{s}}$ and $\omega_{\mathrm{i}}$, respectively, with $\omega_{\mathrm{p}}=\omega_{\mathrm{s}}+\omega_{\mathrm{i}}$. The sample was contained a core Al_xcGa_1-xcAs layer with x_c = 0.17 and a thickness of 230 nm, sandwiched in a six-period Bragg stack made of alternative 127-nm high (Al_0.28Ga_0.72As) and 622-nm low (Al_0.72Ga_0.28As) index layers. The sample was grown along the 001 crystal axis with a width of 5.1 μm and an etching depth of 4.15 μm. The waveguide achieved PM by employing bounded TIR modes and quasi-bounded BRW modes, where TIR modes were commonly formed between high- and low-index claddings, and BRW modes were guided though transverse Bragg reflections at the interface between core and period claddings (see Supplementary Materials 1 for a theoretical analysis and a detailed characterization of the BRW source). The source was pumped by a tunable CW laser, producing degenerate entangled photons centered at 1560.16 nm, which is aligned with the International Telecommunication Union's grids. The polarization entanglement fidelity is above 96% within a 42-nm bandwidth (see Supplementary Materials 2). The temperature of the waveguide can be stabilized by a temperature controller. The wavelength allocation module was realized either with a WSS (model Finisar 16000s) or a coarse WDM unit followed by multiple DWDM filters with 100-/200-GHz spacing, depending on the spectral region of interest. Each detection module consists of an InGaAs avalanche SPD, which has 8.5%/10% detection efficiency, 800-Hz dark count rates, and a 17-μs dead time.

Supplementary Materials Supplementary data to this article can be found online at https://doi.org/10.1016/j.chip.2024.100083.

Acknowledgements We thank Dr. F. Appas for instructive suggestions, and Dr. X.M. Gu for feedback on this manuscript.We acknowledge discussions with Dr. W.C. Ma on entanglement negativity.

Funding This research is supported by the National Natural Science Foundation of China (Grant No. 12274233, 12174187, 62288101). Cheng Qian acknowledges financial support from the Postgraduate Research & Practice Innovation Program of Jiangsu Province (SJCX23_0569).

Declaration of competing interest The authors declare no competing interests.