  Quantum transition probabilities and quantum entanglement for two-qubit
states of a four level trapped ion quantum system are computed for
time-evolving ionic states driven by Jaynes-Cummings Hamiltonians with
interactions mapped onto a $\mbox{SU}(2)\otimes \mbox{SU}(2)$ group structure.
Using the correspondence of the method of simulating a $3+1$ dimensional
Dirac-like Hamiltonian for bi-spinor particles into a single trapped ion, one
preliminarily obtains the analytical tools for describing ionic state
transition probabilities as a typical quantum oscillation feature. For
Dirac-like structures driven by generalized Poincar\'e classes of coupling
potentials, one also identifies the $\mbox{SU}(2)\otimes \mbox{SU}(2)$ internal
degrees of freedom corresponding to intrinsic parity and spin polarization as
an adaptive platform for computing the quantum entanglement between the
internal quantum subsystems which define two-qubit ionic states. The obtained
quantum correlational content is then translated into the quantum entanglement
of two-qubit ionic states with quantum numbers related to the total angular
momentum and to its projection onto the direction of the trapping magnetic
field. Experimentally, the controllable parameters simulated by ion traps can
be mapped into a Dirac-like system in the presence of an electrostatic field
which, in this case, is associated to ionic carrier interactions. Besides
exhibiting a complete analytical profile for ionic quantum transitions and
quantum entanglement, our results indicate that carrier interactions actively
drive an overall suppression of the quantum entanglement.
