This paper explores the connections between particle scattering and quantum information theory in the context of the non-relativistic, elastic scattering of two spin-1/2 particles. An untangled, pure, two-particle in-state is evolved by an S-matrix that respects certain symmetries and the entanglement of the pure out-state is measured. The analysis is phrased in terms of unitary, irreducible representations (UIRs) of the symmetry group in question, either the rotation group for the spin degrees of freedom or the Galilean group for non-relativistic particles. Entanglement may occurs when multiple UIRs appear in the direct sum decomposition of the direct product in-state, but it also depends of the scattering phase shifts.
When two non-relativistic particles scatter in one dimension, they can become entangled. This entanglement process is constrained by the symmetries of the scattering system and the boundary conditions on the incoming state. Applying these constraints, three different mechanisms of entanglement can be identified: the superposition of reflected and transmitted modes, momentum correlations of the reflected mode due to inversion of the relative momentum, and momentum correlations in the transmitted and reflected modes due to dependence of the scattering amplitude on the relative momentum. We consider three standard potentials, the hard core, Dirac delta, and double Dirac delta, and show that the relative importance of these mechanisms depends on the interaction and on the properties of the incoming wavefunction. We find that even when the momenta distributions of the incoming articles are sharply peaked, entanglement due to the momentum correlations generated by reflection can be quite large for particles with unequal mass.
In this paper, a constructive definition of entanglement for nonidentical particles is given. It is shown that particles arising in the process of inelastic scattering are entangled with each other. This property is a consequence of the symmetry of fundamental interactions. Considering detectors of elementary particles, it was concluded that only inelastic scattering can be the basis of a measuring instrument. In the process of inelastic scattering, the measuring instrument can acquire information about the particle (system), while in the case of elastic scattering, it cannot. The nonlinearity of inelastic scattering naturally solves the measurement problem in quantum mechanics.
We consider a general quantum field relativistic scattering involving two half-spin fermions,
A and
B, which are initially entangled with another fermion
C that does not participate in the scattering dynamics. We construct general expressions for the reduced spin matrices for the out-state considering a general tripartite spin-entangled state. In particular we study an inelastic QED process at tree-level, namely ${e}^{\ensuremath{-}}{e}^{+}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{\ensuremath{-}}{\ensuremath{\mu}}^{+}$ and a half-spin fermion
C as a spectator particle which can be entangled to the
AB system in the following ways: W state, GHZ state, $|{\mathrm{A}}^{\ensuremath{\alpha}}⟩\ensuremath{\bigotimes}|{\mathrm{\ensuremath{\Psi}}}^{\ifmmode\pm\else\textpm\fi{}}{⟩}_{\mathrm{BC}}$, and $|{\mathrm{A}}^{\ensuremath{\alpha}}⟩\ensuremath{\bigotimes}|{\mathrm{\ensuremath{\Phi}}}^{\ifmmode\pm\else\textpm\fi{}}{⟩}_{\mathrm{BC}}$, where ${|{\mathrm{\ensuremath{\Psi}}}^{\ifmmode\pm\else\textpm\fi{}}⟩,|{\mathrm{\ensuremath{\Phi}}}^{\ifmmode\pm\else\textpm\fi{}}⟩}$ are the Bell basis states and $|{\mathrm{A}}^{\ensuremath{\alpha}}⟩$ is a spin superposition state of system
A. We calculate the von Neumann entropy variation before and after the scattering for the particle
C and show that spin measurements in
C contain numerical information about the total cross section of the process. We compare the initial states W and GHZ as well as study the role played by the parameter $\ensuremath{\alpha}$ in the evaluation of the entropy variations and the cross section encoded in the spectator particle.