Quantum Entanglement without Spin-Analyzing Power Dependence at the Colliders

We study the quantum entanglement at the colliders which is independent of the spin-analyzing powers. Taking $\mathrm{\Lambda }(\to p{\pi }^{-})\bar{\mathrm{\Lambda }}(\to \bar{p}{\pi }^{+})$ as an example, we investigate whether quantum entanglement in fermion pairs produced at colliders can be certified by using only angular information from final-state decays, while remaining independent of the parity-violating decay parameters ${\alpha }_{\mathrm{\Lambda }}$ and ${\alpha }_{\bar{\mathrm{\Lambda }}}$. Building on a general decomposition of any angular observable in terms of Wigner d-functions, we show that the expectation value must take the form ${\mathcal{O}}_0+{\mathcal{O}}_1{\alpha }_{\mathrm{\Lambda }}+{\mathcal{O}}_2{\alpha }_{\bar{\mathrm{\Lambda }}}+{\mathcal{O}}_3{\alpha }_{\mathrm{\Lambda }}{\alpha }_{\bar{\mathrm{\Lambda }}}$, with coefficients ${\mathcal{O}}_i$ ($i=0,1,2,3$) linear in the spin-density matrix elements ${\alpha }_{k,j}{\alpha }_{m,n}^{\ast }$. We obtain the value ranges of observables over the general and separable spaces of ${\alpha }_{k,j}$, and demonstrate a sufficient entanglement condition for pure states, extending it to mixed states by convexity. In constructing an ${\alpha }_{\mathrm{\Lambda }}$- and ${\alpha }_{\bar{\mathrm{\Lambda }}}$-independent witness from angular observables alone, we find that there are obstacles to probe quantum entanglement via the inequality-type and ratio-type ways. Finally, we present the successful constructions with additional spin information: for the process of $e^{+}{e}^{-}\to J\mathrm{/}\mathrm{\Psi }\to \mathrm{\Lambda }\bar{\mathrm{\Lambda }}$ at $e^{+}{e}^{-}$ collider, independent spin information provided by beam-axis selection enables the construction of normalized observables $f_i(i=1,2)$ that are insensitive to ${\alpha }_{\mathrm{\Lambda }}$ and ${\alpha }_{\bar{\mathrm{\Lambda }}}$; if their measured values lie in $[-1,-\frac{1}{2})\cup (\frac{1}{2},1]$, entanglement is certified, irrespective of purity or mixedness.