Optimizing Entanglement and Bell Inequality Violation in Top Anti-Top Events

TL;DR

This paper formalizes $t\bar t$ spins as a bipartite quantum state and shows that while genuine entanglement and Bell violation are basis- and frame-invariant, the practice of using event-dependent bases yields fictitious states whose entanglement and Bell signals can be optimized. It proves that the diagonal basis, defined as the eigenbasis of the spin-correlation matrix, maximizes both entanglement (via concurrence) and Bell-inequality violation for these fictitious states. The authors analyze QCD sub-processes and compute the optimal basis for the LHC and future $e^+e^-$ colliders, demonstrating that near threshold the diagonal basis aligns with the fixed beam basis while at high energies it aligns with the helicity basis, with significant improvement in Bell-violation sensitivity in realistic setups. They also generalize the approach to asymmetric correlation matrices and CP-violating scenarios, arguing that the method provides a robust, broadly applicable framework for quantum information studies in collider environments and beyond the Standard Model.

Abstract

A top quark and an anti-top quark produced together at colliders have correlated spins. These spins constitute a quantum state that can exhibit entanglement and violate Bell's inequality. In realistic collider experiments, most analyses allow the axes, as well the Lorentz frame to vary event-by-event, thus introducing a dependence on the choice of event-dependent basis leading us to adopt "fictitious states," rather than genuine quantum states. The basis dependence of fictitious states allows for an optimization procedure, which makes the usage of fictitious states advantageous in measuring entanglement and Bell inequality violation. In this work, we show analytically that the basis which diagonalizes the spin-spin correlations is optimal for maximizing spin correlations, entanglement, and Bell inequality violation. We show that the optimal basis is approximately the same as the fixed beam basis (or the rotated beam basis) near the $t\bar t$ production threshold, while it approaches the helicity basis far above threshold. Using this basis, we present the sensitivity for entanglement and Bell inequality violation in $t\bar t$ events at the LHC and a future $e^+e^-$ collider. Since observing Bell inequality violation appears to be quite challenging experimentally, and requires a large dataset in collider experiments, choosing the optimal basis is crucially important to observe Bell inequality violation. Our method and general approach are equally applicable to other systems beyond $t \bar t$, including interactions beyond the Standard Model.

Figures (11)

• Figure 1: Illustration of the spins in a $t\bar{t}$ event produced at the LHC measured (a) in the lab frame and (b) the center-of-mass frame. The two frames are related by a Lorentz boost along the beamline direction. The light blue arrows denote the moving directions of $t$ ($\bar{t}$) in the corresponding frame, and the hollow arrows represent the purposely chosen spin directions of $t$ ($\bar{t}$) in its rest frame.
• Figure 2: (a) Fixed beam basis $(\oldvec{\,x},\oldvec{\,y},\oldvec{\,z})$. (b) Rotated beam basis $(\oldvec{\,x}',\oldvec{\,y}',\oldvec{\,z})$. (c) Helicity basis $(\oldvec{\,r},\oldvec{\,n},\oldvec{\,k})$. (d) Diagonal basis $(\oldvec{\,e}_1^{\,\rm diag},\oldvec{\,e}_2^{\,\rm diag},\oldvec{\,e}_3^{\,\rm diag})$. The rotation angle $\xi$ of the diagonal basis depends on the production process, e.g., $\tan\xi=(1/\gamma)\tan\theta$ for $q\bar{q}\to t\bar{t}$Mahlon:1997uc.
• Figure 3: Spin configurations of $t\bar{t}$ produced from unlike-helicity initial states: (a) for $q\bar{q}\to t\bar{t}$ near threshold, with the cross section proportional to $\beta$; and (b) for $q\bar{q},\ g_L g_R\to t\bar{t}$ in the boosted region. Figure adapted from Ref. Mahlon:2010gw.
• Figure 4: Spin configurations of $t\bar{t}$ produced from like-helicity gluons near and above threshold. The cross section is proportional to $\beta$. Figure adapted from Ref. Mahlon:2010gw.
• Figure 5: Bell inequality violation of angular-averaged states as a function of center-of-mass energy. Bell inequality violation occurs when $\mathscr{B}[\rho]>2$.