Three-Qubit Entangled Embeddings of CPT and Dirac Groups within E 8 Weyl Group

Abstract

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.