Clauser-Horne-Shimony-Holt Bell-inequality Violability with the Full Poincaré-Bloch Sphere

TL;DR

This work extends CHSH Bell tests to the full polarization space by performing elliptical projections on the Poincaré-Bloch sphere, beyond the traditional linear (hd) basis. The authors analyze three polarization-state families (hd linear, hr elliptical, dr elliptical) and mixed hdhr configurations, deriving explicit correlation expressions and mapping the achievable $|S|$ values. Experimentally, polarization-entangled photon pairs are generated with crossed BBO crystals and measured across diverse basis combinations, confirming that Bell states can reach the Tsirelson bound $|S|=2\sqrt{2}$ in suitable bases and that certain non-Bell or mixed-basis configurations also reveal strong nonlocal correlations. The results provide a geometric framework for optimizing CHSH violations, highlight the role of basis choice in revealing hidden nonlocality, and connect Bell nonlocality to steering and entanglement in a more general measurement setting.

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.

Figures (15)

• Figure 1: Scheme of a splitter for any state of polarization $\hbox{$|{ a}\rangle$}$ and the state orthogonal to it $\hbox{$|{ a_\perp}\rangle$}$. It consists of a quarter-wave plate (QWP), half-wave plate (HWP) and a polarizing beam splitter (PBS).
• Figure 2: (a) Bloch sphere for the space of single qubits, where $\hbox{$|{ \pm}\rangle$}=2^{-1/2}(\hbox{$|{ 0}\rangle$}\pm\hbox{$|{ 1}\rangle$})$ and $\hbox{$|{ \pm i}\rangle$}=2^{-1/2}(\hbox{$|{ 0}\rangle$}\pm i\hbox{$|{ 1}\rangle$})$. (b) Poincaré-Bloch sphere, where all states of polarization are represented by a unique point on the sphere. We define the polarization states: $\hbox{$|{ D}\rangle$}=2^{-1/2}(\hbox{$|{ H}\rangle$}+\hbox{$|{ V}\rangle$})$, $\hbox{$|{ A}\rangle$}=2^{-1/2}(\hbox{$|{ H}\rangle$}-\hbox{$|{ V}\rangle$})$, $\hbox{$|{ L}\rangle$}=2^{-1/2}(\hbox{$|{ H}\rangle$}+i\hbox{$|{ V}\rangle$})$, and $\hbox{$|{ R}\rangle$}=2^{-1/2}(\hbox{$|{ H}\rangle$}-i\hbox{$|{ V}\rangle$})$.
• Figure 3: (a) The locus of linear states on the Poincaré-Bloch sphere lie along the great circle that contains states $\hbox{$|{ H}\rangle$}$ and $\hbox{$|{ D}\rangle$}$, and so we call it the "$hd$" great circle. (b) States in this basis are reached by a rotation about the $y$ (R-L) axis.
• Figure 4: (a) The locus of elliptical states on the Poincaré-Bloch sphere with ellipse semi-axes along the horizontal-vertical orientations. (b) States are moved along the $hr$ circle via rotations about the $x$-axis.
• Figure 5: (a) A locus of elliptical states on the Poincaré-Bloch sphere with axes along the diagonal-antidiagonal orientations. (b) states along the $dr$ great circle are transformed by a rotation about the $z$-axis, as illustrated.