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Advanced Photonics Research

Experimental Test of the Three‐Setting Clauser–Horne–Shimony–Holt Inequality with Nonmaximally Orbital Angular Momentum Entangled States

Dongkai Zhang, Xiaodong Qiu, Lixiang Chen

Published Jul 15, 2025• 0 citations

Abstract

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.

DOI: 10.1002/adpr.202500105

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Many experiments have been made to test the Clauser–Horne–Shimony–Holt (CHSH)–Bell inequality (CBI). However, the Cirel’son’s limit has not been reached yet, due to various practically existing imperfections. In this paper, we experimentally investigate how to select the measurement settings for obtaining the optimal violation of CBI for a mixed entangled state. With the commercialized entangled source, we first determine the entanglement degree of the generated photon pairs by using the tomographic technique, and then numerically optimize the measurement settings. Finally, we experimentally obtain the optimal violation of the CBI with the present commercialized entangled source without the maximal entanglement.

Abstract In addition to Bell's original inequality, John Clauser, Michael Horne, Abner Shimony, and Richard Holt (CHSH) developed a special form of Bell's inequality which gives classical limits to the expected correlation between two particles. We present a theoretical demonstration of the maximum value for violation of CHSH inequality, S = 2

\sqrt{2}

, for the pure entangled quantum state. Using the spontaneous four‐wave mixing process, we generate polarization‐entangled photon pairs in a highly nonlinear fiber (HNLF) loop, for experimental verification. Using a nonentangled quantum state, we show theoretically that the violation of CHSH inequality is only a property of entangled states. We also show the theoretical solution of violation of the CHSH inequality for a general Greenberger–Horne–Zeilinger state and for an arbitrary state of two particles. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:2454–2461, 2012; View this article online at wileyonlinelibrary. com. DOI 10.1002/mop.27127