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Three-qubit entangled embeddings of \(CPT\) and Dirac groups within \(E_8\) Weyl group. (English) Zbl 1190.81029
Summary: In quantum information context, the groups generated by Pauli spin matrices, and Dirac gamma matrices, are known as the single qubit Pauli group \({\mathcal P}\), and two-qubit Pauli group \({\mathcal P}_2\), respectively. It has been found [M. Socolovsky, Int. J. Theor. Phys. 43, No. 9, 1941–1967 (2004; Zbl 1073.81033)] that the \(CPT\) group of the Dirac equation is isomorphic to \({\mathcal P}\). 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 \(W(E_8)\) is generated [M. Planat, “Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates”, preprint, arxiv:0904.3691 (2009)]. In this paper, one derives three-qubit entangling groups \(\widetilde{\mathcal P}\) and \(\widetilde{\mathcal P}_2\), isomorphic to the \(CPT\) group \({\mathcal P}\) and to the Dirac group \({\mathcal P}_2\), that are embedded into \(W(E_8)\). One discovers a new class of pure three-qubit 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.

MSC:
81P68 Quantum computation
81P40 Quantum coherence, entanglement, quantum correlations
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
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