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Flavor entanglement in pion-pion scattering at one-loop in chiral perturbation theory

Victor Miguel Banda Guzmán, Rubén Flores Mendieta, Johann Hernández

TL;DR

The paper investigates how flavor entanglement arises in pion-pion scattering at one-loop order in chiral perturbation theory, quantifying it with the quantum mutual information S(ρA,ρB) based on von Neumann entropies. Using the one-loop CHPT amplitude A(s,t,u) derived from the quantum effective action, it shows that entanglement depends on whether the initial pions share identical or distinct flavors, with near-maximal entanglement at threshold: S_MI ≈ 2.8 for identical flavors (three-level qutrit-like system) and S_MI = 2 for distinct flavors (two-level qubit-like system). The authors provide analytical expressions for the entropies S1 and S2, including next-to-leading-order barS1 ≈ 1.4 and barS2 ≈ 1 at threshold using physical Mπ and Fπ, and they discuss the behavior across kinematic regimes via density plots. The work demonstrates that the strong interaction can generate substantial flavor entanglement in scattering and lays out a framework for extending the analysis to higher orders and broader hadronic processes.

Abstract

One of the defining features of quantum mechanics is the ability of two systems to exhibit correlation of information even when they are spatially separated by large distances. This phenomenon, known as quantum entanglement, is a fundamental resource for quantum computation and emerging quantum technologies. Understanding the mechanisms responsible for the generation of entanglement in particle interactions requires a quantitative description of the entanglement produced by fundamental interactions. For the strong interaction, this work presents an analysis of the flavor entanglement generated in pion-pion scattering at one-loop order in chiral perturbation theory. The amount of entanglement is quantified using the quantum mutual information, expressed in terms of the von Neumann entropy. The calculation is performed under fixed pion momenta and initial flavor indices, with the incoming pions assumed to be in a pure, unentangled state. It is shown that the resulting flavor entanglement depends on whether the initial pion flavors are identical or distinct. At threshold, the quantum mutual information reaches approximately 2.8 in the identical-flavor case, close to the maximal value for a pair of qutrits, and equals 2 in the distinct-flavor case, corresponding to the maximal entanglement of two qubits. Analytical expressions for the mutual information at other kinematical regimes are also provided.

Flavor entanglement in pion-pion scattering at one-loop in chiral perturbation theory

TL;DR

The paper investigates how flavor entanglement arises in pion-pion scattering at one-loop order in chiral perturbation theory, quantifying it with the quantum mutual information S(ρA,ρB) based on von Neumann entropies. Using the one-loop CHPT amplitude A(s,t,u) derived from the quantum effective action, it shows that entanglement depends on whether the initial pions share identical or distinct flavors, with near-maximal entanglement at threshold: S_MI ≈ 2.8 for identical flavors (three-level qutrit-like system) and S_MI = 2 for distinct flavors (two-level qubit-like system). The authors provide analytical expressions for the entropies S1 and S2, including next-to-leading-order barS1 ≈ 1.4 and barS2 ≈ 1 at threshold using physical Mπ and Fπ, and they discuss the behavior across kinematic regimes via density plots. The work demonstrates that the strong interaction can generate substantial flavor entanglement in scattering and lays out a framework for extending the analysis to higher orders and broader hadronic processes.

Abstract

One of the defining features of quantum mechanics is the ability of two systems to exhibit correlation of information even when they are spatially separated by large distances. This phenomenon, known as quantum entanglement, is a fundamental resource for quantum computation and emerging quantum technologies. Understanding the mechanisms responsible for the generation of entanglement in particle interactions requires a quantitative description of the entanglement produced by fundamental interactions. For the strong interaction, this work presents an analysis of the flavor entanglement generated in pion-pion scattering at one-loop order in chiral perturbation theory. The amount of entanglement is quantified using the quantum mutual information, expressed in terms of the von Neumann entropy. The calculation is performed under fixed pion momenta and initial flavor indices, with the incoming pions assumed to be in a pure, unentangled state. It is shown that the resulting flavor entanglement depends on whether the initial pion flavors are identical or distinct. At threshold, the quantum mutual information reaches approximately 2.8 in the identical-flavor case, close to the maximal value for a pair of qutrits, and equals 2 in the distinct-flavor case, corresponding to the maximal entanglement of two qubits. Analytical expressions for the mutual information at other kinematical regimes are also provided.

Paper Structure

This paper contains 13 sections, 117 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Chiral perturbation theory at order $\mathcal{O}(p^4)$
  3. The nonlinear realization of the group $\mathrm{SU}(2)\times \mathrm{SU}(2)$
  4. Effective Lagrangian at order $\mathcal{O}(p^4)$
  5. Pion-pion scattering
  6. Pion-pion scattering at one-loop order
  7. Flavor entanglement in the pion-pion scattering process
  8. Conclusions
  9. Transformation properties of the external fields $l^\mu(x)$ and $r^\mu(x)$
  10. Fourier transform identity for the 4-point functional derivative of the Lagrangian term $\mathcal{L}_2^{4\phi}$
  11. One-loop integrals
  12. One-loop corrections to the pion mass and to the residue of the exact pion propagator
  13. Flavor reduced density matrices

Figures (1)

  • Figure 1: Density plots for the entropies $\bar{S}_1$ (left) and $\bar{S}_2$ (right) defined in Eq. \ref{['barS12_oneloop']} as functions of the Maldestam variable $s$ and the scattering angle $\theta$ in the center-of-mass frame.