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

Abstract

We study, as a concrete case study using the $\hat{I}›(\to p\ddot{I}{€}^{-})\bar{\hat{I}}›(\to \bar{p}\ddot{I}{€}^{+})$ 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 $\hat{I}{\pm }_{\hat{I}}›$ and $\hat{I}{\pm }_{\bar{\hat{I}}›}$. 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 ${\mathcal{O}}_0+{\mathcal{O}}_1\hat{I}{\pm }_{\hat{I}}›+{\mathcal{O}}_2\hat{I}{\pm }_{\bar{\hat{I}}›}+{\mathcal{O}}_3\hat{I}{\pm }_{\hat{I}}›\hat{I}{\pm }_{\bar{\hat{I}}›}$, with coefficients ${\mathcal{O}}_i$ ($i=0,1,2,3$) linear in the spin-density matrix elements $\hat{I}{\pm }_{k,j}\hat{I}{\pm }_{m,n}^{\ast }$. We obtain the value ranges of observables over the general and separable spaces of $\hat{I}{\pm }_{k,j}$, and demonstrate a sufficient entanglement condition for pure states, extending it to mixed states by convexity. In constructing an $\hat{I}{\pm }_{\hat{I}}›$- and $\hat{I}{\pm }_{\bar{\hat{I}}›}$-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 $\hat{I}{\pm }_{k,j}(k,j=\pm \frac{1}{2})$ 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

Cited by in corpus (1)

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