Determining the Spin Density Matrix via Its Rank and Probing the Quantum Entanglement and Bell Non-Locality at the Lepton Colliders

Abstract

Considering two-fermion $F_a F_b$ productions and decays via one scalar or photon exchange at the $e^+e^-$ collider, we show that the rank $r_{\rho}$ of spin density matrix $\rho$ is equal to the number of degrees of freedom of the mediator. For one generic scalar exchange, the spin density matrix is rank one for a pure state. With rank-one condition, we can determine the spin analyzing powers for $F_a$ and $F_b$ and their product if the CP symmetry is violated and conserved, respectively, and probe the CP violation. These results can be applied to the $\eta_c \to \Lambda {\bar \Lambda}$ at the BESIII experiment and the Higgs $\to \tau \tau $ at the LHC. For one photon exchange, the spin density matrix is rank two for a mixed state. Considering the $\Lambda \bar \Lambda $ productions and decays at the BESIII experiment as an example, we show that the spin analyzing powers for $F_a$ and $ F_b$ can be determined by the rank-two conditions in details. Therefore, we can reconstruct the spin density matrices, probe the quantum entanglement and Bell non-locality, and evade the no-go theorem. Furthermore, we conjecture that the $N\times N$ spin density matrix with $r_{\rho} \le N-2 $ can be reconstructed at the lepton colliders in general.