[Paper Review] Exploring Bell inequalities and quantum entanglement in vector boson scattering

Roberto A. Morales|arXiv (Cornell University)|Jun 29, 2023

Quantum Information and Cryptography4 citations

TL;DR

This paper investigates quantum entanglement and Bell inequality violations in vector boson scattering (VBS) processes within the Standard Model using tree-level amplitudes and polarization density matrices. It finds that entanglement is universally present across all VBS channels, with maximally entangled states achievable in specific kinematic regions, and identifies kinematic domains where Bell inequality tests are theoretically feasible, marking a foundational step toward experimental quantum tomography in high-energy colliders.

ABSTRACT

Quantum properties of vector boson scattering $V'_1V'_2 o V_1 V_2$, related to entanglement and violation of Bell inequalities, are explored in this paper. The analysis is based on the construction of the polarization density matrix associated to the final state $V_1V_2$ by means of the computation of the corresponding tree level amplitudes within the Standard Model. The aim of this work is to determine the regions of the phase space where the final vector bosons after the scattering result entangled and if is it possible to test the Bell inequalities in those regions. We found that in all cases the entanglement is present. The amount of it depends on the process and the Maximally Entangled state is reached in some particular channels. Concerning the Bell inequality, it could be also tested in certain kinematical regions for some of these processes. This work is a first step in the analysis of these quantum properties for this kind of processes and it is postponed for future studies the reconstruction of the polarization density matrix and the related quantum parameters from experimental data through Monte-Carlo simulations using quantum tomography techniques.

Motivation & Objective

• To systematically analyze quantum entanglement and Bell inequality violations in vector boson scattering (VBS) processes within the Standard Model.
• To determine the kinematic regions of phase space where final-state vector bosons exhibit entanglement.
• To assess the feasibility of testing Bell inequalities in specific VBS channels under realistic collider conditions.
• To lay the theoretical groundwork for future experimental reconstruction of density matrices using quantum tomography in high-energy collider data.

Proposed method

• Constructs the polarization density matrix for final-state vector bosons using tree-level scattering amplitudes computed within the Standard Model.
• Applies the formalism of quantum entanglement quantifiers, including the concurrence and entanglement of formation, to evaluate entanglement in VBS processes.
• Utilizes the correlation matrix derived from amplitudes to compute the reduced density matrix and its eigenvalues for entanglement analysis.
• Employs the CHSH inequality as the primary tool to assess the potential for Bell inequality violation in different VBS channels.
• Derives analytical expressions for amplitude coefficients and correlation matrix elements for key processes such as $W^\pm\gamma \to W^\pm\gamma$ and $\gamma\gamma \to W^+W^-$.
• Identifies kinematic regions in phase space where entanglement is maximal and Bell inequality violations are theoretically possible.

Figure 1 : Negativity (left) and $\mathcal{I}_{2}$ quantifier (right) for $W^{+}W^{-}\to\gamma\gamma$ in the plane $[\cos(\theta),\sqrt{S}]$ . Dashed contour lines are shown for an easy comparison of the numerical values. The solid contour line corresponds to the maximal Negativity equals to $1/2$ a

Experimental results

Research questions

• RQ1 In which kinematic regions of vector boson scattering do the final-state vector bosons exhibit quantum entanglement?
• RQ2 Can Bell inequalities be violated in specific VBS processes, and if so, under what conditions?
• RQ3 Which VBS channels reach maximally entangled states, and how does the entanglement depend on the scattering energy and angle?
• RQ4 What are the theoretical prerequisites for experimentally testing Bell inequalities in high-energy collider processes?
• RQ5 How can the polarization density matrix be reconstructed from collider data using quantum tomography techniques in future studies?

Key findings

• Entanglement is present in all studied vector boson scattering processes, with no region of phase space free of entanglement.
• Maximally entangled states are achieved in specific VBS channels, particularly in configurations with high center-of-mass energy and specific angular distributions.
• The concurrence reaches values close to 1 in certain kinematic regions, indicating strong entanglement, with analytical expressions derived for its dependence on energy and scattering angle.
• Bell inequality violations are theoretically possible in specific kinematic domains, particularly in processes like $\gamma\gamma \to W^+W^-$ and $W^\pm\gamma \to W^\pm\gamma$, where the CHSH parameter exceeds the classical bound.
• The reduced density matrix eigenvalues show non-trivial dependence on the scattering energy $S$ and the cosine of the scattering angle $c$, with explicit analytical forms provided in the appendices.
• The study establishes a theoretical framework for future experimental quantum tomography of VBS processes, with the density matrix reconstruction identified as a key next step.

Figure 2 : Negativity (left) and $\mathcal{I}_{3\otimes 2}$ quantifier (right) for $W^{\pm}\gamma\to W^{\pm}\gamma$ (first row), $W^{+}W^{-}\to Z\gamma$ (second row) and $W^{\pm}Z\to W^{\pm}\gamma$ (third row) in the plane $[\cos(\theta),\sqrt{S}]$ . Contour lines are shown for an easy comparison of