We study, as a concrete case study using the \( \Lambda \left(\to p{\pi}^{-}\right)\overline{\Lambda}\left(\to \overline{p}{\pi}^{+}\right) \) system, whether quantum entanglement in fermion pairs produced at colliders can be certified solely using angular information from final-state decays, while remaining independent of the parity-violating decay parameters αΛ and \( {\alpha}_{\overline{\Lambda}} \) . 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 \( {\mathcal{O}}_0+{\mathcal{O}}_1{\alpha}_{\Lambda}+{\mathcal{O}}_2{\alpha}_{\overline{\Lambda}}+{\mathcal{O}}_3{\alpha}_{\Lambda}{\alpha}_{\overline{\Lambda}} \) , with coefficients 𝒪i (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 αk,j, and demonstrate a sufficient entanglement condition for pure states, extending it to mixed states by convexity. In constructing an αΛ- and \( {\alpha}_{\overline{\Lambda}} \) -independent witness from angular observables alone, we find that there are obstacles to probe quantum entanglement via the inequality-type and ratio-type ways. In particular, for the ratio-type criterion 〈A〉/〈B〉, the presence of zeros of 〈B〉 in both the general and separable spaces of \( {\alpha}_{k,j}\left(k,j=\pm \frac{1}{2}\right) \) results in identical value ranges of 〈A〉/〈B〉 in the two spaces (covering the entire real line), thereby precluding any effective criterion. Finally, for this specific system, we present the successful constructions with additional spin information: for the process of \( {e}^{+}{e}^{-}\to J/\Psi \to \Lambda \overline{\Lambda} \) at an e+e− collider, independent spin information provided by beam-axis selection enables the construction of normalized observables fi (i = 1, 2) that are insensitive to αΛ and \( {\alpha}_{\overline{\Lambda}} \) ; if their measured values lie in \( \left[\left.-1,-\frac{1}{2}\right)\cup \left(\frac{1}{2},1\right.\right] \) , entanglement is certified, irrespective of purity or mixedness.
