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AVS Quantum Science

Clauser–Horne–Shimony–Holt Bell-inequality violability with the full Poincaré–Bloch sphere

Carlos Cardoso-Isidoro, Enrique J. Galvez

Published Apr 28, 2026• 0 citations
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Abstract

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.

DOI
10.1116/5.0320174

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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.
This work presents a method to establish entanglement of photon pairs in mixed (linear and elliptical) polarisation states. For such configurations, the standard Clauser–Horne–Shimony–Holt (CHSH) inequalities are projected to fail.Exploiting trigonometrical symmetries, we define a geometric correlation parameter S†S^\daggerS†, inspired by CHSH-type methods but algebraically distinct, which maintains maximal violation at 2\sqrt{2}2​ for all elliptical polarisation distortions induced by quarter-wave plates, and remains invariant across all four Bell states—∣ψ±⟩\ket{\psi^\pm}∣ψ±⟩ and ∣ϕ±⟩\ket{\phi^\pm}∣ϕ±⟩. This result contrasts sharply with local hidden variable (LHV) models, which remain bounded by S†≤1.5S^\dagger \leq 1.5S†≤1.5 under the same conditions. Numerical simulations confirm that the proposed test functions as a robust entanglement witness; experimental verification is now essential.
The Clauser-Horne-Shimony-Holt inequality (CHSH) is one of the most popular and well-studied witnesses of Bell's theorem, separating classical from quantum correlations. In this work, for every $d\ensuremath{\ge}2$, we present a generalization of the CHSH inequality with the property that maximal violation is achieved uniquely by the maximally entangled state of local dimension ddd. This provides an avenue for device-independent certification of maximally entangled states of arbitrary local dimension.
We theoretically and experimentally investigate the optimal conditions for the Bell experiment using spontaneous parametric down-conversion (SPDC) sources. In theory, we show that a relatively large average photon number (typically $\ensuremath{\sim}0.5$) is desirable to observe the maximum violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. Moreover, we show that the violation converges to 0.27 in the limit of large average photon number. In experiment, we perform the Bell experiment without postselection using polarization-entangled photon pairs at 1550-nm telecommunication wavelength generated from SPDC sources. While the violation of the CHSH inequality is not directly observed due to the overall detection efficiencies of our system, the experimental values agree well with those obtained by the theory with experimental imperfections. Furthermore, in the range of small average photon numbers ($\ensuremath{\le}0.1$), we propose and demonstrate a method to estimate the ideal CHSH value intrinsically contained in the tested state from the lossy experimental data without assuming the input quantum state.
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.