Three-Body Non-Locality in Particle Decays

TL;DR

This work analyzes three-body decays $X\to A B C$ into three massless spin-$\tfrac{1}{2}$ fermions to explore tripartite entanglement and Bell nonlocality within a general four-fermion interaction framework. By applying the Mermin and tight $4\times4\times2$ inequalities, the authors map how scalar, vector, and tensor Lorentz structures generate distinct entanglement patterns and nonlocal correlations across the decay phase space, highlighting when fully local-real and bipartite local-real descriptions fail. Scalar interactions yield bi-separable states with no genuine tripartite nonlocality, vector interactions can produce genuine tripartite entanglement and detectable nonlocality in certain regions, and tensor interactions can saturate quantum bounds and exhibit GHZ-like correlations. The study provides a detailed methodology for optimizing measurement axes and demonstrates that no single Bell inequality universally detects all non-FLR correlations, underscoring the need for multiple, tailored observables in experimental tests of quantum foundations in high-energy processes.

Abstract

The exploration of entanglement and Bell non-locality among multi-particle quantum systems offers a profound avenue for testing and understanding the limits of quantum mechanics and local real hidden variable theories. In this work, we examine non-local correlations among three massless spin-1/2 particles generated from the three-body decay of a massive particle, utilizing a framework based on general four-fermion interactions. By analyzing several inequalities, we address the detection of deviations from quantum mechanics as well as violations of two key hidden variable theories: fully local-real and bipartite local-real theories. Our approach encompasses the standard Mermin inequality and the tight $4 \times 4 \times 2$ inequality, providing a comprehensive framework for probing three-partite non-local correlations. Our findings provide deeper insights into the boundaries of classical and quantum theories in three-particle systems, advancing the understanding of non-locality in particle decays and its relevance to particle physics and quantum foundations.

Figures (7)

• Figure 1: The concurrence triangle and a genuine tripartite entanglement measure $F_3$.
• Figure 2: Different types of non-localities and their detections in the three-qubit system. It is seen that the 442 inequality functional with the value 4 is tight, that is, it describes the boundary of the FLR polytope.
• Figure 3: The momentum and spin configuration. The momentum of A is fixed to the $z$-direction. At the rest frame of X, the decay plane is aligned with the $x$-$z$ plane and the two opening angles, A-B and A-C, are given by $\theta_B$ and $\theta_C$, respectively, with $0 \le \theta_B, \theta_C \le \pi$ and $\pi \le \theta_B + \theta_C \le 2 \pi$. The spin direction of X is given by ${\bf n}=(\sin \theta \cos\phi, \sin \theta \sin\phi, \cos\theta)$.
• Figure 4: The dependence of various entanglement measures, $F_3$ (black dotted), ${\cal C}_{A(BC)}$ (magenta dotted) and ${\cal C}_{B(AC)} = {\cal C}_{C(AB)}$ (purple dashed-dotted), and the Bell-type observables, ${\langle {\cal B}_{\rm M} \rangle}/2$ (blue solid), ${\langle {\cal B}_{\rm S} \rangle}/4$ (green solid) and $\langle {\cal B}^{\rm sym}_{442} \rangle/4$ (red solid), on the initial spin direction ${\bf n}$, assuming the vector interaction in Eq. \ref{['Lvector']}. The maximum values of the corresponding local-real theories normalise the values of Bell-type observables. In the left and right columns, the decay angles are fixed at $(\theta_B, \theta_C) = (\frac{4 \pi}{6}, \frac{5 \pi}{6})$ and $(\frac{2 \pi}{6}, \frac{5 \pi}{6})$, respectively. In the upper (lower) panels, the initial spin is rotated about the $y$ ($x$) axis clockwise, and the horizontal axis indicates the rotation angle $\omega_y$ ($\omega_x$).
• Figure 5: The dependence of the GTE measure, $F_3$, (the first line), $\langle {\cal B}_{\rm M} \rangle$ (the second line), $\langle {\cal B}_{\rm S} \rangle$ (the third line) and $\langle {\cal B}^{\rm sym}_{442} \rangle$ (the fourth line) on the decay angles $\theta_B$ and $\theta_C$, evaluated assuming the vector interaction Eq. \ref{['Lvector']}. In the left, middle and right columns, the initial spin is fixed as ${\bf n} \propto {\bf e}_z$, ${\bf e}_y$ and $({\bf e}_x + {\bf e}_y + {\bf e}_z)$, respectively. For the Bell-type observables, the colour indicates the expectation values of the observables normalised by their local-real theory maxima. Namely, the colour indicates $\langle {\cal B}_{\rm M} \rangle/2$, $\langle {\cal B}_{\rm S} \rangle/4$ and $\langle {\cal B}^{\rm sym}_{442} \rangle/4$ for the plots in the second, third and fourth lines, respectively. The maximum of the colour bar is set at the corresponding quantum mechanical maximum values, $F_3^{\rm max} = 1$, $\langle {\cal B}_{\rm M} \rangle_{\rm QM}^{\rm max} = 4$, $\langle {\cal B}_{\rm S} \rangle_{\rm QM}^{\rm max} = 4\sqrt{2}$ and $\langle {\cal B}^{\rm sym}_{442} \rangle_{\rm QM}^{\rm max} = 8$. The black solid curves appearing in some of the $\langle {\cal B}_{\rm M} \rangle$ and $\langle {\cal B}_{\rm S} \rangle$ plots indicate the saturation of the corresponding local-real bounds, $\langle {\cal B}_{\rm M} \rangle = 2$ and $\langle {\cal B}_{\rm S} \rangle = 4$.