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Experimental test of the chained CHSH inequality in a two-dimensional orbital angular momentum system

Dongkai Zhang, Xiaodong Qiu, Lixiang Chen

Published Feb 12, 2025• 1 citation
Publisher

Abstract

The chained CHSH inequality is of paramount importance in quantum information science, demonstrating superiority over the CHSH inequality in some quantum information tasks. Hitherto, experimental verification was rarely considered to verify the chained CHSH inequality, as previous experimental efforts primarily focused on the CHSH inequality. Here, by employing two-photon orbital angular momentum (OAM) entanglement, we experimentally demonstrate the chained CHSH across multiple settings. In theory, we demonstrate a geometrical connection of optimal measurement vectors and establish that the optimal measurement vectors reside in a circle through the center of the Bloch sphere. This allows us to construct multiple sets of optimal measurement vectors on different circles. In the experiment, we typically choose circles in the XOY, XOZ, and YOZ planes to observe violations of the chained CHSH inequality. Our experimental results indicate that the maximum violation increases with setting k , underscoring the dependence of the maximum violation on the measurement angles.

DOI
10.1364/oe.546423

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The chained CHSH inequality is of paramount importance in quantum information science, demonstrating superiority over the CHSH inequality in some quantum information tasks. Hitherto, experimental verification was rarely considered to verify the chained CHSH inequality, as previous experimental efforts primarily focused on the CHSH inequality. Here, by employing two-photon orbital angular momentum (OAM) entanglement, we experimentally demonstrate the chained CHSH across multiple settings. In theory, we demonstrate a geometrical connection of optimal measurement vectors and establish that the optimal measurement vectors reside in a circle through the center of the Bloch sphere. This allows us to construct multiple sets of optimal measurement vectors on different circles. In the experiment, we typically choose circles in the XOY, XOZ, and YOZ planes to observe violations of the chained CHSH inequality. Our experimental results indicate that the maximum violation increases with setting k , underscoring the dependence of the maximum violation on the measurement angles.
Nonmaximally entangled states are pivotal for quantum technologies, offering enhanced noise resilience and resource efficiency for quantum communication tasks. However, the maximum violation of the multisetting Clauser–Horne–Shimony–Holt (CHSH) inequality in nonmaximally entangled systems has yet to be explored, primarily due to the algebraic complexity associated with such systems. Herein, it is demonstrated that the maximization of the correlation measure in the three‐setting CHSH inequality, for both pure and mixed states, can be elegantly reformulated as a geometric optimization problem: maximizing the perimeter of a triangle inscribed within an ellipse. Leveraging this analytical insight, the maximum violation of the three‐setting CHSH inequality using photonic orbital angular momentum entanglement and spatial light modulators for precise intensity measurements is experimentally realized, observing that the Bell parameter diminishes to the classical bound as entanglement decreases. The findings shed new light on optimal measurement strategies for nonmaximally entangled states, demonstrating conclusively that phase‐only measurements are insufficient for achieving maximum violations in such states. Thus, the work may offer valuable guidance for harnessing nonmaximally entangled states in practical quantum technologies.
Characterizing high-dimensional entangled states is of crucial importance in quantum information science and technology. Recent theoretical progress has been made to extend Hardy's paradox into a general scenario with multisetting and multidimensional systems, which enables the maximum probability of nonlocal events to surpass the bound that was limited in Hardy's original test. Hitherto, no experimental verification has been conducted to verify Hardy's paradox, as most of the previous experimental efforts were restricted to two-dimensional systems. Here, based on two-photon high-dimensional orbital angular momentum (OAM) entanglement, we report an experiment to demonstrate Hardy's paradox for multiple settings and multiple outcomes. We demonstrate the paradox for two-setting higher-dimensional OAM subspaces up to d=7d=7d=7, which reveals that the nonlocal events increase with dimension. Furthermore, we showcase the nonlocality with an experimentally recorded probability of 36.77% for a five-setting three-dimensional OAM subspace via entanglement concentration, thus showing a sharper contradiction between quantum mechanics and classical theory.
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.
The symmetric orbital angular momentum (OAM) entanglement in the subspace of { ± ℓ} has been extensively exploited in quantum information science. Here we investigate instead the Bell inequalities for the two-dimensional subspace spanned by asymmetric OAM modes of ℓ 1 and ℓ 2 , usually corresponding to the quantum property of nonmaximal entanglement. We demonstrate, both theoretically and experimentally, the optimal violation of a suitable Clauser-Horne-Shimony-Holt Bell inequality for the asymmetric OAM modes, thus manifesting the entanglement for asymmetric OAM states space.
We report a violation of the CHSH inequality for ghost-images. This is achieved by using two spatially separated phase modulators within the context of a two-photon parametric down-conversion experiment. We obtain edge enhanced images as a direct consequence of the quantum correlations in the orbital angular momentum (OAM) of the down-converted photon pairs. For phase objects, with differently orientated edges, we show a violation of the CHSH Bell-type inequality for an OAM subspace, thereby unambiguously revealing the quantum nature of our ghost-imaging arrangement.
We use spatial light modulators (SLMs) to measure correlations between arbitrary superpositions of orbital angular momentum (OAM) states generated by spontaneous parametric down-conversion. Our technique allows us to fully access a two-dimensional OAM subspace described by a Bloch sphere, within the higher-dimensional OAM Hilbert space. We quantify the entanglement through violations of a Bell-type inequality for pairs of modal superpositions that lie on equatorial, polar, and arbitrary great circles of the Bloch sphere. Our work shows that SLMs can be used to measure arbitrary spatial states with a fidelity sufficient for appropriate quantum information processing systems.