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IEEE Transactions on Quantum Engineering

Effect of Nonidealities on Nonlocal Quantum Entanglement Quality as Measured by the CHSH Inequality Violation

R. A. Brewster, J. Goldhar, Mark B. Morris, Gerald Baumgartner, Yanne K. Chembo

Published Jan 1, 2026• 0 citations
Publisher

Abstract

The violation of Bell-like inequalities unambiguously verifies the entanglement of a multipartite quantum state. However, experimental impairments and nonidealities often compromise the entanglement quality, and this degradation translates into a nonmaximal violation of these inequalities. We present a detailed analysis of the influence of experimental impairments on the quality of nonlocal polarization entanglement for a two-qubit state as measured by the Clauser–Horne–Shimony–Holt parameter, with a particular emphasis on beamsplitters, polarization controllers, filters, and photon sources. This analysis allows the identification of the impairments that are critical and that need to be specifically monitored at the experimental level.

DOI
10.1109/tqe.2026.3732924

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The violation of Bell-like inequalities unambiguously verifies the entanglement of a multipartite quantum state. However, experimental impairments and nonidealities often compromise the entanglement quality, and this degradation translates into a nonmaximal violation of these inequalities. We present a detailed analysis of the influence of experimental impairments on the quality of nonlocal polarization entanglement for a two-qubit state as measured by the Clauser–Horne–Shimony–Holt parameter, with a particular emphasis on beamsplitters, polarization controllers, filters, and photon sources. This analysis allows the identification of the impairments that are critical and that need to be specifically monitored at the experimental level.
It is well-known that in certain scenarios weakly entangled states can generate stronger nonlocal effects than their maximally entangled counterparts. In this paper, we consider violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality when one party has inefficient detectors, a scenario known as an asymmetric Bell experiment. For any fixed detection efficiency, we derive a simple upper bound on the entanglement needed to violate the inequality by more than some specified amount $\ensuremath{\kappa}\ensuremath{\ge}0$. When $\ensuremath{\kappa}=0$, the amount of entanglement in all states violating the inequality goes to zero as the detection efficiency approaches 5050%50 from above. We finally consider the scenario in which detection inefficiency arises for only one choice of local measurement. In this case, it is shown that the CHSH inequality can always be violated for any nonzero detection efficiency and any choice of noncommuting measurements.
Abstract The Clauser–Horne–Shimony–Holt (CHSH) inequality test is widely used as a mean of invalidating the local deterministic theories. Most attempts to experimentally test nonlocality have presumed unphysical idealizations that do not hold in real experiments, namely, noiseless measurements. We demonstrate an experimental violation of the CHSH inequality that is free of idealization and rules out local models with high confidence. We show that the CHSH inequality can always be violated for any nonzero noise parameter of the measurement. Intriguingly, less entanglement exhibits more nonlocality in the CHSH test with noisy measurements. Furthermore, we theoretically propose and experimentally demonstrate how the CHSH test with noisy measurements can be used to detect weak entanglement on two-qubit states. Our results offer a deeper insight into the relation between entanglement and nonlocality.
Linearly polarized projections are the tacit means for performing Clauser-Horne-Shimony-Holt (CHSH) Bell-inequality tests using polarization-entangled photon pairs. The inequality is valid for all states on the Poincaré-Bloch sphere, but few laboratory studies have investigated violations with the full sphere. In this article, we explore the experimental verifications of the predicted violations of the CHSH inequality with Bell and non-Bell states with same and different linear and elliptically polarized basis states for each photon. We find that Bell states violate CHSH when using the same basis for both photons, regardless of their ellipticity, whereas they show no violations for photon projections in different bases. We found non-Bell maximally-entangled states for which the converse is true.
Linearly polarized projections are the tacit means for performing Clauser–Horne–Shimony–Holt (CHSH) Bell-inequality tests using polarization-entangled photon pairs. The inequality is valid for all states on the Poincaré–Bloch sphere, but few laboratory studies have investigated violations with the full sphere. In this article, we explore the experimental verifications of the predicted violations of the CHSH inequality with Bell and non-Bell states with same and different linear and elliptically polarized basis states for each photon. We find that Bell states violate CHSH when using the same basis for both photons, regardless of their ellipticity, whereas they show no violations for photon projections in different bases. We found non-Bell maximally entangled states for which the converse is true.
The experimental test of Bell’s inequality is mainly focused on Clauser–Horne–Shimony–Holt (CHSH) form, which provides a quantitative bound, while little attention has been paid to the violation of Wigner inequality (WI). Based on the spin coherent state quantum probability statistics, we in the present paper extend the WI and its violation to arbitrary two-spin entangled states with antiparallel and parallel spin-polarizations. The local part of density operator gives rise to the WI while the violation is a direct result of nonlocal interference between two components of the entangled states. The Wigner measuring outcome correlation denoted by [Formula: see text] is always less than or at most equal to zero for the local realist model ([Formula: see text]) regardless of the specific initial state. On the other hand, the violation of WI is characterized by any positive value of [Formula: see text], which possesses a maximum violation bound [Formula: see text] [Formula: see text]. We conclude that the WI is equally convenient for the experimental test of violation by the quantum entanglement.