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Optics Continuum

Optimal violation of a Bell inequality in two-dimensional state spaces for an asymmetric orbital angular momentum system

Sijin Yan, Dongkai Zhang, Lixiang Chen

Published Oct 18, 2024• 1 citation
Publisher

Abstract

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.

DOI
10.1364/optcon.540458

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Orbital angular momentum (OAM) entanglement is investigated in the Bessel-Gaussian (BG) basis. Having a readily adjustable radial scale, BG modes provide an alternative basis for OAM entanglement over Laguerre-Gaussian modes. We show that the OAM bandwidth in terms of BG modes can be increased by selection of particular radial wavevectors and leads to a flattening of the spectrum, which allows for higher dimensionality in the entangled state. We demonstrate entanglement in terms of BG modes by performing a Bell-type experiment and showing a violation of the Clauser-Horne-Shimony-Holt inequality for the ℓ = ±1 subspace. In addition, we use quantum state tomography to indicate higher-dimensional entanglement in terms of BG modes.
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.
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.
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.