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arXiv:2601.02350 · Physical Review Letters

Binarisation-loophole-free observation of high-dimensional quantum nonlocality

Jia-le Miao, Elna Svegborn, Zhuo Chen, Yu Guo, Xiao-Min Hu, Yun-Feng Huang, Chuan-Feng Li, Guang-Can Guo, Armin Tavakoli, Bi-Heng Liu

quant-phSubmitted Jan 5, 2026• Announced Jan 6, 2026• 0 citations
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Abstract

Bell inequality tests based on high-dimensional entanglement usually require measurements that can resolve multiple possible outcomes. However, the implementation of high-dimensional multi-outcome measurements is often only emulated via a collection of ``click or no-click'' measurements. This reduction of multi-outcome measurements to binary-outcome measurements opens a loophole in high-dimensional tests Bell inequalities which can be exploited by local hidden variable models [Tavakoli et al., Phys. Rev. A 111, 042433 (2025)]. Here, we close this loophole by using four-dimensional photonic path-mode entanglement and multi-outcome detection. We test both the well-known Collins-Gisin-Linden-Massar-Popescu inequality and a related Bell inequality tailored for maximally entangled states in high-dimension. We observe violations that are large enough to also rule out any quantum model based on entanglement of lower dimension, thereby demonstrating genuinely high-dimensional nonlocality free of the binarisation loophole.

Journal reference
Physical Review Letters 136, 240201 (2026)
DOI
10.1103/ydch-vgmy
Comments
6 pages, 4 figures. Supplemental material included

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Bell inequality tests based on high-dimensional entanglement usually require measurements that can resolve multiple possible outcomes. However, the implementation of high-dimensional multi-outcome measurements is often only emulated via a collection of ``click or no-click'' measurements. This reduction of multi-outcome measurements to binary-outcome measurements opens a loophole in high-dimensional tests Bell inequalities which can be exploited by local hidden variable models [Tavakoli et al., Phys. Rev. A 111, 042433 (2025)]. Here, we close this loophole by using four-dimensional photonic path-mode entanglement and multi-outcome detection. We test both the well-known Collins-Gisin-Linden-Massar-Popescu inequality and a related Bell inequality tailored for maximally entangled states in high-dimension. We observe violations that are large enough to also rule out any quantum model based on entanglement of lower dimension, thereby demonstrating genuinely high-dimensional nonlocality free of the binarisation loophole.
Abstract Violation of Bell inequalities is an essential requirement for many quantum information and communication protocols. In high-dimensional systems, Bell inequality tests face the challenge of implementing genuinely multi-outcome measurements, since the emulation of these with separate dichotomic projections opens a binarisation loophole that local hidden variable theories can exploit. Here we show that the joint spectral intensity of a two-photon entangled state contains access to the necessary multi-outcome measurements to overcome this obstacle and certify nonlocality for high-dimensional states. This result is contrary to the belief that the joint spectral intensity is a phase-insensitive quantity and does not have sufficient information to certify entanglement or nonlocality. Using this approach, we violate the CGLMP Bell inequality up to dimension d = 8, all with negligible p-values, and for the first time close the binarisation loophole in high-dimensional Bell experiments. Guaranteeing nonlocal correlations using frequency-only measurements removes the technological hurdle of measurements in the temporal domain, thus greatly simplifying any practical implementation of future high-dimensional quantum information protocols.
Violation of Bell inequalities is an essential requirement for many quantum information and communication protocols. In high-dimensional systems, Bell inequality tests face the challenge of implementing genuinely multi-outcome measurements, since the emulation of these with separate dichotomic projections opens a binarisation loophole that local hidden variable theories can exploit. Here we show that the joint spectral intensity of a two-photon entangled state contains access to the necessary multi-outcome measurements to overcome this obstacle and certify and violate a Bell inequality for high-dimensional states. This result is contrary to the belief that the joint spectral intensity is a phase-insensitive quantity and does not have sufficient information to certify entanglement or Bell-nonlocality. Using this approach, we violate the CGLMP Bell inequality up to dimension d = 8, all with negligible p-values, and for the first time close the binarisation loophole in high-dimensional Bell experiments. Guaranteeing Bell-nonlocal correlations using frequency-only measurements removes the technological hurdle of measurements in the temporal domain, thus greatly simplifying any practical implementation of future high-dimensional quantum information protocols.
Violation of Bell's inequalities shows strong conflict between quantum mechanics and local realism. Loophole-free Bell tests not only deepen understanding of quantum mechanics, but are also important foundations for device-independent (DI) tasks in quantum information. High-dimensional quantum systems offer a significant advantage over qubits for closing the detection loophole. In the symmetric scenario, a detection efficiency as low as 61.8% can be tolerated using four-dimensional states and a four-setting Bell inequality [Phys. Rev. Lett. 104, 060401 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.060401]. For the first time, we show that four-dimensional entangled photons violate a Bell inequality while closing the detection loophole in experiment. The detection efficiency of the four-dimensional entangled source is about 71.7%, and the fidelity of the state is 0.995±0.001. Combining the technique of multicore fibers, the realization of loophole-free high-dimensional Bell tests and high-dimensional quantum DI technologies are promising.
We present a source of entangled photons that violates a Bell inequality free of the "fair-sampling" assumption, by over 7 standard deviations. This violation is the first reported experiment with photons to close the detection loophole, and we demonstrate enough "efficiency" overhead to eventually perform a fully loophole-free test of local realism. The entanglement quality is verified by maximally violating additional Bell tests, testing the upper limit of quantum correlations. Finally, we use the source to generate "device-independent" private quantum random numbers at rates over 4 orders of magnitude beyond previous experiments.
Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest (2,2,2)(2,2,2)(2,2,2) scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the (2,2,3)(2,2,3)(2,2,3) scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.
Characterizing high-dimensional entangled states holds pivotal significance in quantum information science and technology. Recent theoretical progress has been made to extend the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality into a general scenario with multisetting and multidimensional systems. Here, by employing two-photon orbital angular momentum entanglement, we conduct an experiment to demonstrate the CGLMP inequality across multiple settings and outcomes. Our experimental results violate the CGLMP inequality by more than 12 standard deviations, demonstrating that the maximum violation increases with both dimension and setting. Furthermore, we reveal, both theoretically and experimentally, the logical connection between the CGLMP inequality and the general Hardy's paradox. Intriguingly, our analysis establishes that, even in high-dimensional systems, Hardy's paradox can be regarded as a special instance of Bell inequality, with their respective bounds aligning under the no-signaling principle and the information causality principle. Our work may advance the foundational understanding of quantum correlations in high-dimensional quantum systems.