Bypassing Spin-Analyzing Power Dependence for Quantum Entanglement at Colliders: A Case Study of $Λ\barΛ$

Abstract

We study, as a concrete case study using the $Λ(\to pπ^-)\barΛ(\to \bar{p}π^+)$ 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 $α_{\barΛ}$. 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 $\mathcal{O}_0+\mathcal{O}_1α_Λ+\mathcal{O}_2α_{\barΛ}+\mathcal{O}_3α_Λα_{\barΛ}$, with coefficients $\mathcal{O}_i$ ($i=0,1,2,3$) linear in the spin-density matrix elements $α_{k,j}α^*_{m,n}$. 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 $α_{\barΛ}$-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 ${\langle A\rangle}/{\langle B\rangle}$, the presence of zeros of $\langle B\rangle$ in both the general and separable spaces of $α_{k,j}(k,j=\pm\frac{1}{2})$ results in identical value ranges of ${\langle A\rangle}/{\langle B\rangle}$ 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.