Quantum Tomography of Fermion Pairs in $e^+e^-$ Collisions: Longitudinal Beam Polarization Effects

TL;DR

This work develops a uniform quantum tomography framework for two-qubit fermion systems produced in $e^+e^-$ collisions and studies how longitudinal beam polarization controls entanglement, Bell nonlocality, and magic across $t\bar{t}$, $\mu^+\mu^-$, and $e^+e^-$ final states. Using a Fano-Bloch decomposition, it quantifies $\mathcal{C}$, $\mathcal{B}$, and $\mathcal{M}_2$ and reveals distinct polarization dependences: $s$-channel–dominated processes largely reshape single-spin polarizations and $\mathcal{M}_2$, while $\mathcal{C}$ and $\mathcal{B}$ remain robust; Bhabha scattering shows pronounced $s$/$t$-channel interference, enabling richer polarization control of all observables. The paper provides realistic observability projections for future colliders, identifying optimal polarization configurations and angular regions to achieve $>5\sigma$ significance for all three quantum resources, thereby enabling experimental exploration of quantum-information concepts in high-energy leptonic collisions. These results illuminate the practical potential of quantum resources as probes of Standard Model dynamics and offer guidance for incorporating quantum-information measurements into collider programs.

Abstract

We present a quantum tomography study of fermion pair production at future $e^+e^-$ colliders, emphasizing how longitudinal beam polarization controls the two-qubit spin density matrix. We study the processes $e^+ e^- \to t\bar{t},\ e^+e^-\to μ^+μ^-$ and Bhabha scattering $e^+e^-\to e^+e^-$, representing the mass threshold behavior, the $Z$ pole resonance and the $s/t$-channel interplay. We choose to focus on three key concepts: quantum entanglement via the concurrence $\mathcal{C}$, Bell nonlocality via the optimal Clauser Horne Shimony Holt (CHSH) parameter $\mathcal{B}$, and non-stabilizerness (``magic'') via the second stabilizer Rényi entropy $\mathcal{M}_2$. For the $s$-channel-dominated channels, longitudinal polarization mainly reshapes single-spin polarizations while leaving the spin-correlation matrix largely unchanged, rendering $\mathcal{C}$ and $\mathcal{B}$ comparatively robust, but inducing a pronounced variation of $\mathcal{M}_2$. In contrast, in Bhabha scattering, polarization modifies the relative contributions of the $s$-channel and $t$-channel and can strongly affect all three observables. The observability of entanglement, Bell nonlocality, and magic exceeds the $5σ$ level when both statistical and systematic uncertainties are included, establishing the fermion pair systems as ideal laboratories for quantum-information studies in high energy leptonic collisions. With optimized beam polarization, future $e^+e^-$ colliders will provide a unique opportunity to experimentally explore and influence quantum resources in particle interactions.

Figures (33)

• Figure 1: Definition of the helicity basis $\{\hat{r},\,\hat{n},\,\hat{k}\}$ in the $f\bar{f}$ center-of-mass frame. The unit vector $\hat{k}$ points along the fermion momentum, $\hat{n}$ is normal to the scattering plane, and $\hat{r}=\hat{n}\times\hat{k}$ completes a right-handed triad (equivalently $\hat{n}=\hat{k}\times\hat{r}$).
• Figure 2: Coefficients $a$ and $b$ in Eq. (\ref{['eq:psi']}) for the quantum state of the $t\bar{t}$ system for $RL$ ($LR$) beam polarizations as a function of the scattering angle $\theta$ at a velocity $\beta \to 1$.
• Figure 3: Contour plot for purity of the $t\bar{t}$ final state at an $e^+ e^-$ collider, shown in the kinematic plane of scattering angle $\theta$ and c. m. energy $\sqrt{s}$ in GeV.
• Figure 4: Concurrence $\mathcal{C}$ (left) and Bell variable $\mathcal{B}$ (right) for $e^+e^-\to t\bar{t}$ as a function of scattering angle $\theta$ at $\sqrt{s}=$ 360 GeV (red) and 1 TeV (blue), with different beam polarizations: unpolarized (solid lines), fully polarized $e^-_L e^+_R$ (dashed lines), and $e^-_R e^+_L$ (dotted lines).
• Figure 5: Contour plots for the concurrence (left) and the Bell variable (right) in the plane of $\theta-\sqrt s$ (GeV) for $e_L^-e_R^+\to t\bar{t}$.