Quantum spin correlations in $Z^\prime$-mediated $t\bar{t}$ production at future lepton colliders

We study quantum spin correlations in top-quark pair production at future lepton colliders in the presence of a neutral gauge boson from anomaly-free general $U(1)$ extensions of the Standard Model. The process $\ell^+\ell^-\to t\bar t$, with $\ell=e,\mu$, is analyzed through the spin-density matrix including $\gamma$, $Z$ and $Z^\prime$ exchange and their interference. We focus on quantum-information observables such as the sufficient entanglement marker $\mathcal{D}_{\min}$, concurrence, purity and the maximal Clauser-Horne-Shimony-Holt (CHSH) parameter, and compare their behavior with conventional rate information. Within the $U(1)_X$ framework, we consider several representative charge assignments to investigate how different chiral structures influence these observables, with particular emphasis on the $Z^\prime$ resonance region and polarized $e^-e^+$ collisions, where the two allowed initial-state helicity configurations can be selectively enhanced. We show that electron-beam polarization provides a direct handle on the left- and right-handed lepton charges of various $U(1)_X$ scenarios. These results demonstrate that quantum spin observables provide information complementary to cross sections and angular distributions in searches for chiral neutral gauge interactions.