  Quantum entanglement is analyzed thoroughly in the case of the ground and
lowest states of two-electron axially symmetric quantum dots under a
perpendicular magnetic field. The individual-particle and the center-of-mass
representations are used to study the entanglement variation at the transition
from interacting to noninteracting particle regimes. The mechanism of symmetry
breaking due to the interaction, that results in the states with symmetries
related to the later representation only, being entangled even at the vanishing
interaction, is discussed. The analytical expression for the entanglement
measure based on the linear entropy is derived in the limit of noninteracting
electrons. It reproduces remarkably well the numerical results for the lowest
states with the magnetic quantum number M>2 in the interacting regime. It is
found that the entanglement of the ground state is a discontinuous function of
the field strength. A method to estimate the entanglement of the ground state,
characterized by the quantum number M, with the aid of the magnetic field
dependence of the addition energy is proposed.
