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arXiv:2305.10027 · Canadian Journal of Physics

Entanglement entropy of Compton scattering with a witness

Shanmuka Shivashankara

Published Jul 31, 2023• 14 citations
arXivHTMLPDF

Abstract

Unitarity and the optical theorem are used to derive the reduced density matrices of Compton scattering in the presence of a witness particle. Two photons are initially entangled wherein one photon participates in Compton scattering, while the other is a witness, i.e., does not interact with the electron. Unitarity is shown to require that the entanglement entropy of the witness photon does not change after its entangled partner undergoes scattering. The final mutual information of the electron's and witness particle’s polarizations is shown to be nonzero for low-energy Compton scattering. This indicates that the two particles became correlated in spite of no direct interaction. Assuming an initial maximally entangled state, the change in entanglement entropy of the scattered photon’s polarization is calculated in terms of Stokes parameters. A common ratio of areas occurs in the final reduced density matrix elements, von Neumann entropies, Stokes parameter, and mutual information. This common ratio consists of the Thomson scattering cross-section and an accessible regularized scattering area.

DOI
10.1139/cjp-2023-0142

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Unitarity and the optical theorem are used to derive the reduced density matrices of Compton scattering in the presence of a witness particle. Two photons are initially entangled wherein one photon participates in Compton scattering, while the other is a witness, i.e., does not interact with the electron. Unitarity is shown to require that the entanglement entropy of the witness photon does not change after its entangled partner undergoes scattering. The final mutual information of the electron's and witness particle’s polarizations is shown to be nonzero for low-energy Compton scattering. This indicates that the two particles became correlated in spite of no direct interaction. Assuming an initial maximally entangled state, the change in entanglement entropy of the scattered photon’s polarization is calculated in terms of Stokes parameters. A common ratio of areas occurs in the final reduced density matrix elements, von Neumann entropies, Stokes parameter, and mutual information. This common ratio consists of the Thomson scattering cross-section and an accessible regularized scattering area.
When two initially entangled photons each undergo Compton scattering, the scattered electrons become correlated. However, the final reduced density matrix of one scattered pair is not influenced by the other scattered pair due to unitarity. Herein, we keep unitarity up to tree level for Compton double scattering and obtain different results than recent literature. The initial four particles, where the initial photons are entangled, are written as a superposition of two states with a relative phase. The final density matrix has two area divergences that are regularized with unitarity. The regularization procedure, i.e. solving for the roots of a polynomial that represents the probability for no scattering, suggests a novel definition of the scattering cross-section. Vieta's formulas relate these divergences to finite cross-sections. For an initial pure state, the formulas for the final density matrix and the correlation of final electronic polarizations are given. The correlation implies double scattering is analogous to Young's diffraction experiment. The two initial superposed states are the circular apertures while the Feynman amplitudes are the interfering complex light fields.
The Compton scattering of an entangled pair of photons is studied. Using the resulting reduced density matrix, we demonstrate how to derive the Kraus operators for the interaction from the underlying fundamental theory, quantum electrodynamics. The normalization of the final density matrix of the entangled photons after the interaction is crucial to obtaining the correct probabilities and correlations. The reduced density matrix of the photon, which does not participate in the interaction directly, remains unchanged. It indicates that quantum effects in quantum sensing using an entangled photon pair can be observed only through the correlations or superposition of the entire pair. We relate our results to the Stokes parameters for determining polarizations and investigating the entangled states of photons from electron-positron annihilation in astrophysical systems.
Regularized quantum information metrics are calculated for the scattering process e − e + → γ , Z → μ − μ + that has a witness photon entangled with the initial electron-positron state. Unitarity implies the correct regularization of divergences that appear in both the final density matrix and von Neumann entanglement entropies. The entropies are found to quantify uncertainty or randomness. The variation of information, entanglement entropy, and correlation between the muon’s and witness photon’s helicities are found to convey equivalent information. The magnitude of the muon’s expected helicity rises (falls) as the helicity entropy falls (rises). Area, or the scattering cross section, is a source of entropy for the muon’s helicity entropy and momentum entropy. The muon’s differential angular entropy distribution is similar to the differential angular cross section distribution, capturing the forward-backward asymmetry at high center-of-mass energies. Published by the American Physical Society 2024
In this paper, the 16 Stokes parameters of a two-photon system are organized into a $4\ifmmode\times\else\texttimes\fi{}4$ Stokes matrix. This formalism is then applied to analyze the polarization properties of annihilation photons produced by electron-positron annihilation, with a particular focus on their evolution during Compton scattering interactions. The study demonstrates that when one of the subsystems undergoes Compton scattering, quantum entanglement preserves the polarization symmetry between the scattered and noninteraction photons, while in the absence of entanglement, this symmetry is lost. Moreover, the method simplifies the calculation of the joint differential cross section for multiple scattering scenarios and provides a means to characterize the polarization properties of the individual subsystems. These findings highlight the impact of quantum correlations on scattering behavior, and they offer a framework for further investigation into the polarization dynamics of entangled photon pairs.
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We study the behavior of entanglement between different degrees of freedom of scattering fermions, based on an exemplary QED scattering process ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$. The variation of entanglement entropy between two fermions from an initial state to the final state was computed, with respect to different entanglement between the ingoing particles. This variation of entanglement entropy is found to be proportional to an area quantity, the total cross section. We also study the spin-momentum and helicity-momentum entanglements within one particle in the aforementioned scattering process. The calculations of the relevant variations of mutual information in the same inertial frame reveals that, for a maximally entangled initial state, the scattering between the particles does not affect the degree of both of these entanglements of one particle in the final state. It is also found that the increasing degree of entanglement between two ingoing particles would restrict the generation of entanglement between spin (helicity) and momentum of one outgoing particle. And the entanglement between spin and momentum within one particle in the final state is shown to always be stronger than that for helicity-momentum for a general initial entanglement state, implying significantly distinct properties of entanglement for the helicity and spin perceived by an inertial observer.