  In quantum information context, the groups generated by Pauli spin matrices,
and Dirac gamma matrices, are known as the single qubit Pauli group P, and
two-qubit Pauli group P2, respectively. It has been found [M. Socolovsky, Int.
J. Theor. Phys. 43, 1941 (2004)] that the CPT group of the Dirac equation is
isomorphic to P. One introduces a two-qubit entangling orthogonal matrix S
basically related to the CPT symmetry. With the aid of the two-qubit swap gate,
the S matrix allows the generation of the three-qubit real Clifford group and,
with the aid of the Toffoli gate, the Weyl group W(E8) is generated (M. Planat,
Preprint 0904.3691). In this paper, one derives three-qubit entangling groups ?
P and ? P2, isomorphic to the CPT group P and to the Dirac group P2, that are
embedded into W(E8). One discovers a new class of pure theequbit quantum states
with no-vanishing concurrence and three-tangle that we name CPT states. States
of the GHZ and CPT families, and also chain-type states, encode the new
representation of the Dirac group and its CPT subgroup.
