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

Abstract: We present a quantum tomography study of fermion pair production at future $e<sup>+e<sup>-$ colliders, emphasizing how longitudinal beam polarization controls the two-qubit spin density matrix. We study the processes $$e⁺ e^- \to t\bar{t},\ e^{+e^-\to} μ<sup>+μ<sup>-$$ and Bhabha scattering $e^{+e^-\to} e<sup>+e<sup>-$, 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 $e<sup>+e<sup>-$ colliders will provide a unique opportunity to experimentally explore and influence quantum resources in particle interactions.