Bell's theorem, demonstrating the incompatibility of quantum predictions with local hidden variable (LHV) models, is a cornerstone of quantum foundations. Loophole-free violations of Bell inequalities have recently been achieved, paving the way for quantum network applications in areas like quantum communication, distributed computing, cryptography, and randomness generation. Device-independent (DI) protocols, relying solely on observed data with minimal assumptions on measurement apparatuses, are particularly promising. However, most work focuses on bipartite scenarios. Multipartite generalizations, often using complex states like Greenberger-Horne-Zeilinger (GHZ) states, present challenges in experimental implementation. Quantum networks for the quantum internet are more likely to comprise independent sources generating smaller entangled states with higher quality and rate. The independence of these sources leads to richer correlations than in standard Bell nonlocality scenarios, offering advantages like nonlocality activation and less stringent detection efficiencies. Experimental implementations have been limited to simple tripartite networks and bilocality models. This paper presents a proof-of-principle demonstration of a scalable approach, extending beyond bilocality by using a star-shaped network with up to four independent sources and five nodes. The researchers aim to violate chained n-locality inequalities, demonstrating nonlocal correlations in a DI manner. This scalable approach opens avenues for exploring diverse network topologies and device-independent information processing protocols.

The paper reviews the history of Bell's theorem and loophole-free Bell inequality violations. It highlights the importance of device-independent protocols in quantum information processing and the limitations of previous work focusing on bipartite scenarios and the complexities of using multipartite entangled states like GHZ states. The authors discuss the advantages of using independent sources in quantum networks, including nonlocality activation and relaxed detection efficiency requirements. The literature on bilocality scenarios and their experimental limitations are examined, setting the stage for the proposed experiment's significance. The causal modeling approach using directed acyclic graphs (DAGs) is introduced as a tool for analyzing complex networks.

The researchers employed a causal modeling approach using DAGs to represent and analyze the star-network topology. The simplest network considered was the bilocality scenario, involving two independent sources and three nodes, where a nonlinear Bell inequality was used to test for non-classical correlations. The study then extended to n-locality scenarios, which involve n independent sources distributing correlations among n+1 nodes in a star network. A chained n-locality inequality was formulated to characterize non-classical correlations within this framework. The experiment utilized a photonic platform with four independent polarization-entangled photon pair sources located in four separate laboratories, ensuring the independence of the sources. Each laboratory housed a source and a measurement station, with one laboratory also hosting the central node. Three sources used spontaneous parametric down-conversion (SPDC) of type II in BBO crystals pumped by different pulsed lasers, while the fourth source used type II SPDC in a periodically poled KTP crystal in a Sagnac interferometer, pumped by a continuous-wave laser. Entangled photon pairs were generated, with one photon sent to the central node and the other measured locally. Polarization analysis was performed using half-wave plates (HWPs) and polarizing beam splitters (PBSs). Coincidence counting was performed across the different laboratories using time-taggers, synchronizing events and filtering noise. The experiments tested networks with n = 2, 3, and 4 sources, with various measurement settings (k) for each node. The optimal quantum violation of the n-locality inequality was calculated, and experimental values were compared against the classical bound, taking noise into account. The measurement settings for peripheral and central nodes were meticulously described.

The experimental results showed clear violations of the chained n-locality inequality for different numbers of sources (n) and measurement settings (k). For k=2, violations were observed with n=2, 3, and 4 sources, exceeding the classical bound by a significant number of standard deviations (109, 50, and 38 respectively). With increasing k values, the experiment demonstrated violations for the 2-, 3-, and 4-source star networks, demonstrating a significant violation even with a narrowed coincidence window (down to 0.49 µs), which reduces the probability of uncontrolled communication between measurement stations. Specifically, for n=4 and k=4, the violation of the inequality surpassed the classical limit by 71 standard deviations, fully compatible with theoretical predictions considering noise. The observed correlations were incompatible with classical n-local models but compatible with a local model without source independence, illustrating the importance of considering source independence to witness quantum nonlocality.

The significant violations of the chained n-locality inequalities demonstrate the presence of non-classical correlations in the star-shaped quantum networks. This is significant as it moves beyond the previously studied bilocality scenarios and establishes the feasibility of exploring more complex network topologies. The use of independent sources in separate laboratories effectively addresses the independence loophole often present in such experiments. While the experiment does not fully close the locality loophole (requiring greater spatial separation of the parties), the use of a narrow coincidence window minimizes the possibility of causal influences between the measurement stations. The high-level of violations observed, coupled with the versatility of the platform, points toward future implementations of multipartite protocols such as secret sharing, where the independence of sources and the simultaneous presence of nonlocal correlations are crucial. The platform’s adaptability makes it suitable for testing diverse network configurations, including triangle networks and linear chain topologies relevant to quantum repeaters.

The paper successfully demonstrates the experimental violation of chained n-locality inequalities in star-shaped quantum networks with up to four independent sources and five nodes. This work showcases a scalable platform for studying nonlocality in complex quantum networks, which is a crucial step toward developing quantum technologies. The versatility of the setup allows for future investigations of different network topologies and multipartite quantum protocols. Further research could focus on closing the locality loophole and exploring applications in device-independent quantum information processing.

The experiment relies on the measurement independence assumption and the assumption of a lack of correlations among the sources, which are difficult to perfectly enforce. The locality loophole is not fully closed due to the lack of space-like separation between measurement stations. Though the narrow coincidence window reduces the likelihood of causal influences, it doesn’t eliminate the possibility entirely. The fair-sampling assumption is also made, as in other relevant experiments. These limitations, though not fatal to the main conclusions, should be considered when interpreting the results and planning for future experiments.