A top quark and an antitop quark produced together at colliders have correlated spins. These spins constitute a quantum state that can exhibit entanglement and violate Bell’s inequality. In realistic collider experiments, most analyses allow the axes, as well the Lorentz frame, to vary event by event, thus introducing a dependence on the choice of event-dependent basis leading us to adopt “fictitious states,” rather than genuine quantum states. The basis dependence of fictitious states allows for an optimization procedure, which makes the usage of fictitious states advantageous in measuring entanglement and Bell inequality violation. In this work, we show analytically that the basis that diagonalizes the spin-spin correlations is optimal for maximizing spin correlations, entanglement, and Bell inequality violation. We show that the optimal basis is approximately the same as the fixed beam basis (or the rotated beam basis) near the t t ¯ production threshold, while it approaches the helicity basis far above threshold. Using this basis, we present the sensitivity for entanglement and Bell inequality violation in t t ¯ events at the Large Hadron Collider (LHC) and a future e + e − collider. Since observing Bell inequality violation appears to be quite challenging experimentally, and requires a large dataset in collider experiments, choosing the optimal basis is crucially important to observe Bell inequality violation. Our method and general approach are equally applicable to other systems beyond t t ¯ , including interactions beyond the Standard Model.
This paper proposes a generalized Bell-like inequality (GBI) for multiparticle entangled Schr\"odinger cat states of arbitrary spin
s. Based on quantum probability statistics, the GBI and violation are formulated in a unified manner with the help of a state density operator, which can be separated into local and nonlocal parts. The local part gives rise to the inequality, while the nonlocal part is responsible for the violation. The GBI is not violated at all by the quantum average, except the spin-
1/2 entangled states. If the measuring outcomes are restricted in the subspace of the spin coherent state (SCS), namely, only the maximum spin values $\ifmmode\pm\else\textpm\fi{}s$, the GBI is still meaningful for the incomplete measurement. With the help of SCS quantum probability statistics, it is proved that the violation of GBI can occur only for half-integer spins and not integer spins. Moreover, the maximum violation bound depends on the number parity of the entangled particles, which is
1/2 for the odd particle numbers and 1 for the even numbers.