Bypassing Spin-Analyzing Power Dependence for Quantum Entanglement at Colliders: A Case Study of $\Lambda \bar{\Lambda }$

Abstract

We study, as a concrete case study using the $\Lambda (\to p{\pi }^{-})\bar{\Lambda }(\to \bar{p}{\pi }^{+})$ 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 ${\alpha }_{\Lambda }$ and ${\alpha }_{\bar{\Lambda }}$. 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 ${\mathcal{O}}_0+{\mathcal{O}}_1{\alpha }_{\Lambda }+{\mathcal{O}}_2{\alpha }_{\bar{\Lambda }}+{\mathcal{O}}_3{\alpha }_{\Lambda }{\alpha }_{\bar{\Lambda }}$, 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 ${\alpha }_{\Lambda }$- and ${\alpha }_{\bar{\Lambda }}$-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 ${\alpha }_{k,j}(k,j=\pm \frac{1}{2})$ 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.

10 pages, 1 table. Published version

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Cited by in corpus (1)

• Excluding Local Hidden Variables in $\Lambda \bar{\Lambda }$ Production: The Incompatibility with Angular-Momentum Conservation and CPT Invariance