Entanglement and Bell Nonlocality in $τ^+ τ^-$ at the BEPC

TL;DR

The paper addresses measuring quantum entanglement and Bell nonlocality in the τ^+τ^- final state produced in high-energy e^+e^- collisions near threshold. It adopts a density-matrix framework for τ^+τ^- and uses two complementary quantum-information approaches—decay-based tomography and kinematic density-matrix reconstruction—to extract entanglement C(ρ) and Bell nonlocality B from collider data, supported by the Fano-Bloch representation. The study shows that the current ψ(2S) dataset at BEPC-II can reveal entanglement if systematic uncertainties are at or below 1%, while future BEPC-II runs in the 4.0–5.6 GeV range can achieve entanglement with ≈4% precision and potentially establish Bell nonlocality at ≥5σ provided systematics are tightly controlled (0.5–2%). This work demonstrates the feasibility of quantum-information measurements in collider environments and provides a quantitative roadmap for BES-III to perform entanglement and Bell tests, with broader implications for exploring additional quantum-information quantities in high-energy physics.

Abstract

Quantum entanglement and Bell nonlocality are two phenomena that occur only in quantum systems. In both cases, these are correlations between two subsystems that are classically absent. Traditionally, these phenomena have been measured in low-energy photon and electron experiments, but more recently they have also been measured in high-energy particle collider environments. In this work, we propose measuring the entanglement and Bell nonlocality in the $τ^+τ^-$ state near and above its kinematic threshold at the Beijing Electron Positron Collider (BEPC). We find that in the existing dataset, entanglement is observable if systematic uncertainties are kept to 1%. In the upcoming run between 4.0 and 5.6 GeV, the entanglement is predicted to be measurable with a precision better than 4% and Bell nonlocality can be established at $5σ$ as long as systematic uncertainty can be controlled at level of 0.5% - 2.0%, depending on the center-of-mass energy.

Figures (7)

• Figure 1: The diagonal basis of the $\tau^-\tau^+$ system near threshold. It is related to the helicity basis by a rotation of $\xi$.
• Figure 2: The energy spectrum (left) and angular distribution (right) of $\pi^\pm$ from the decays $\tau^- \to \nu_\tau \pi^-$ and $\tau^+ \to \bar{\nu}_\tau \pi^+$.
• Figure 3: The concurrence (left) and Bell variable (right) as a function of scattering angle $\theta$ at the center-of-mass energies of $\sqrt{s} = m_{\psi(2S)} = 3.7~{\rm GeV}$, $\sqrt{s} = 4.0~{\rm GeV}$, and $\sqrt{s} = 5.6~{\rm GeV}$.
• Figure 4: The significance of the concurrence (left) and Bell variable (right) with the decay method as a function of the size of the angular window $\Delta\theta$ at the center-of-mass energies of $\sqrt{s} = m_{\psi(2S)} = 3.7~{\rm GeV}$, $\sqrt{s} = 4.0~{\rm GeV}$, and $\sqrt{s} = 5.6~{\rm GeV}$ using the decay method with the $\nu\pi$ decay channel and the integrated luminosity $L=20\ \mathrm{fb^{-1}}$ for $\sqrt{s} = 4.0~{\rm GeV}$ and $\sqrt{s} = 5.6~{\rm GeV}$ and $N_{\psi(2S) \to \tau^+ \tau^-} =3.5\times10^6$ for $\sqrt{s} = m_{\psi(2S)}$. The systematic uncertainty $\Delta_{\rm sys}$ is not included.
• Figure 5: The optimal significance of observing entanglement (left) and Bell inequality violation (right) as a function of integrated luminosity at 5.6 GeV and $4.0-5.6$ GeV using the decay method in the $\nu\pi$ channel and $\Delta_{\rm sys}=2\%$. The angular window cut is optimized at $L=20~{\rm fb}^{-1}$ and used for all luminosity values.