Bypassing spin-analyzing power dependence for quantum entanglement at colliders: a case study of $$ \Lambda \overline{\Lambda} $$

A bstract We study, as a concrete case study using the $$ \Lambda \left(\to p{\pi}^{-}\right)\overline{\Lambda}\left(\to \overline{p}{\pi}^{+}\right) $$ Λ → p π − Λ ¯ → p ¯ π + 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 α Λ and $$ {\alpha}_{\overline{\Lambda}} $$ α Λ ¯ . 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 $$ {\mathcal{O}}_0+{\mathcal{O}}_1{\alpha}_{\Lambda}+{\mathcal{O}}_2{\alpha}_{\overline{\Lambda}}+{\mathcal{O}}_3{\alpha}_{\Lambda}{\alpha}_{\overline{\Lambda}} $$ O 0 + O 1 α Λ + O 2 α Λ ¯ + O 3 α Λ α Λ ¯ , with coefficients 𝒪 i ( i = 0 , 1 , 2 , 3) linear in the spin-density matrix elements $$ {\alpha}_{k,j}{\alpha}_{m,n}^{\ast } $$ α k , j α m , n ∗ . We obtain the value ranges of observables over the general and separable spaces of α k,j , and demonstrate a sufficient entanglement condition for pure states, extending it to mixed states by convexity. In constructing an α Λ - and $$ {\alpha}_{\overline{\Lambda}} $$ α Λ ¯ -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 $$ {\alpha}_{k,j}\left(k,j=\pm \frac{1}{2}\right) $$ α k , j k j = ± 1 2 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 $$ {e}^{+}{e}^{-}\to J/\Psi \to \Lambda \overline{\Lambda} $$ e + e − → J / Ψ → Λ Λ ¯ at an 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 α Λ and $$ {\alpha}_{\overline{\Lambda}} $$ α Λ ¯ ; if their measured values lie in $$ \left[\left.-1,-\frac{1}{2}\right)\cup \left(\frac{1}{2},1\right.\right] $$ − 1 − 1 2 ∪ 1 2 1 , entanglement is certified, irrespective of purity or mixedness.