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arXiv:2407.16305 · Physical Review A

Binarization of multioutcome measurements in high-dimensional quantum correlation experiments

Armin Tavakoli, Roope Uola, Jef Pauwels

Published Apr 30, 2025• 3 citations
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Abstract

High-dimensional systems are an important frontier for photonic quantum correlation experiments. These correlation tests commonly prescribe measurements with many possible outcomes but they are often implemented through many individual binary-outcome measurements that use only a single-detector. Here, we discuss how this binarization procedure for multioutcome measurements can open a loophole, unless specific device-characterization assumptions are satisfied. We highlight that correlation tests designed for multioutcome measurements can be trivialized in binarized implementations and we then show how to accurately analyze binarized data to reveal their quantum features. For seminal types of correlation experiments, such as Bell inequality tests, steering tests and prepare-and-measure experiments, we find that binarization may incur a sizable cost in the magnitude of quantum advantages. This emphasizes the importance of both accurate data analysis and implementing genuinely multioutcome measurements in high-dimensional correlation experiments.

DOI
10.1103/physreva.111.042433

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High-dimensional systems are an important frontier for photonic quantum correlation experiments. These correlation tests commonly prescribe measurements with many possible outcomes but they are often implemented through many individual binary-outcome measurements that use only a single-detector. Here, we discuss how this binarization procedure for multioutcome measurements can open a loophole, unless specific device-characterization assumptions are satisfied. We highlight that correlation tests designed for multioutcome measurements can be trivialized in binarized implementations and we then show how to accurately analyze binarized data to reveal their quantum features. For seminal types of correlation experiments, such as Bell inequality tests, steering tests and prepare-and-measure experiments, we find that binarization may incur a sizable cost in the magnitude of quantum advantages. This emphasizes the importance of both accurate data analysis and implementing genuinely multioutcome measurements in high-dimensional correlation experiments.
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.
Certifying quantum behavior from classically accessible data is essential for secure communication and scalable quantum technologies. While powerful certification methods such as Bell nonlocality and quantum steering exist, their implementation typically requires entanglement or additional assumptions, and experimental demonstrations mainly focus on low-dimensional systems. In minimal prepare-and-measure scenarios, where a sender encodes information into quantum states and a receiver performs a single measurement, robust certification becomes particularly challenging, especially in the presence of noise and in higher-dimensional Hilbert spaces. Here, we propose, design, and experimentally implement a protocol that certifies quantumness between two distant parties without the need for preshared resources or measurement incompatibility. The experiments are carried out using the orbital angular momentum degrees of freedom of single photons, chosen for providing increased dimensionality that is scalable. We demonstrate the robustness of the protocol through rank-stability analysis of the observed correlations, which enables the certification of non-classicality even in the presence of noise. Our results provide a practical route to validate high-dimensional quantum communication systems and open new possibilities for secure and dimension-efficient quantum information processing.
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.
Violating Bell's inequalities (BIs) allows one to certify the preparation of entangled states from minimal assumptions-in a device-independent manner.Finding BIs tailored to many-body correlations as prepared in present-day quantum computers and simulators is however a highly challenging endeavor.In this work, we focus on BIs violated by very coarse-grain features of the system: two-body correlations averaged over all permutations of the parties.For two-outcome measurements, specific BIs of this form have been theoretically and experimentally studied in the past, but it is practically impossible to explicitly test all such BIs.Data-driven methods-reconstructing a violated BI from the data themselves-have therefore been considered.Here, inspired by statistical physics, we develop a novel data-driven approach specifically tailored to such coarse-grain data.Our approach offers two main improvements over the existing literature: (1) it is directly designed for any number of outcomes and settings; (2) the obtained BIs are quadratic in the data, offering a fundamental scaling advantage for the precision required in experiments.This very flexible method, whose complexity does not scale with the system size, allows us to systematically improve over all previously known Bell inequalities robustly violated by ensembles of quantum spin 1/2; and to discover novel families of Bell inequalities, tailored to spin-squeezed states and many-body spin singlets of arbitrary spin-j ensembles.