Quantum Entanglement without Spin-Analyzing Power Dependence at the Colliders

TL;DR

This work investigates whether quantum entanglement in collider-produced fermion pairs, exemplified by Λ–¯Λ, can be certified using only angular information from weak-decay final states while remaining independent of the spin-analyzing powers α_Λ and α_{¯Λ}. It develops a Wigner-d-function–based formalism that yields ⟨O⟩ = 𝒪0 + 𝒪1 α_Λ + 𝒪2 α_{¯Λ} + 𝒪3 α_Λ α_{¯Λ}, with coefficients 𝒪i linear in the spin-density matrix elements, and defines value ranges R1 and R2 for the general and separable α-spaces. The paper shows obstacles to α-independent, angle-only entanglement witnesses for both inequality-type and ratio-type criteria and demonstrates that, without extra spin information, angle-based tests can fail to distinguish entangled from separable states. It then presents a viable route by incorporating beam-axis spin information in e^+e^− → J/ψ → ΛΛ̄, building normalized observables f1,f2 that are independent of α_Λ and α_{¯Λ}, and establishing entanglement certification when f_i lie in the specified ranges. Overall, the results delineate the limits of angle-only QE certification at colliders and offer practical guidance for leveraging external spin information to realize robust entanglement witnesses in hyperon systems.

Abstract

We study the quantum entanglement at the colliders which is independent of the spin-analyzing powers. Taking $Λ(\to pπ^-)\barΛ(\to \bar{p}π^+)$ 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 $α_Λ$ 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. Finally, we present the successful constructions with additional spin information: for the process of $e^+e^-\to J/Ψ\to Λ\barΛ$ 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 $α_Λ$ and $α_{\barΛ}$; if their measured values lie in $\left[-1,-\tfrac{1}{2}\right)\cup\left(\tfrac{1}{2},1\right]$, entanglement is certified, irrespective of purity or mixedness.