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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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References:
 [1] Socolovsky, M.: The CPT group of the Dirac field. Int. J. Theor. Phys. 43, 2004 (1941) · Zbl 1073.81033 [2] Planat, M.: Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates. (2009). Preprint 0904.3691 [quant-ph] · Zbl 1225.81038 [3] Mermin, N.D.: Hidden variables and the two theorems of John Bell. Rev. Mod. Phys. 65, 803 (1993) · doi:10.1103/RevModPhys.65.803 [4] Kauffman, L.H., Lomonaco, S.J.: Braiding operators are universal quantum gates. New J. Phys. 6, 134 (2004) · doi:10.1088/1367-2630/6/1/134 [5] Planat, M., Jorrand, P.: Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces. J. Phys. A: Math. Theor. 41, 182001 (2008) · Zbl 1139.81022 · doi:10.1088/1751-8113/41/18/182001 [6] Varlamov, V.V.: The CPT group in the de Sitter space. Ann. Fond. Louis Broglie 29(2), 969 (2004) · Zbl 1062.81085 [7] Nebe, G., Rains, E.M., Sloane, N.J.A.: The invariants of the Clifford groups. Des. Codes Cryptogr. 24, 99 (2001) · Zbl 1002.11057 · doi:10.1023/A:1011233615437 [8] Coffman, V., Kundu, J., Wootters, W.K.: Distributed entanglement. Phys. Rev. A 61, 052306 (2000) · doi:10.1103/PhysRevA.61.052306 [9] Dür, W., Vidal, G., Cirac, J.J.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000) · doi:10.1146/annurev.physiol.62.1.1 [10] Lohmayer, R., Osterloh, A., Siewert, J., Uhlman, A.: Entangled three-qubit states without concurrence and three-tangle. Phys. Rev. Lett. 97, 260502 (2006) · Zbl 1228.81071 · doi:10.1103/PhysRevLett.97.260502 [11] Gross, D., Audenaert, K., Eisert, J.: Evenly distributed unitaries: on the structure of unitary designs. J. Math. Phys. 48, 052104 (2007) · Zbl 1144.81351 · doi:10.1063/1.2716992 [12] Weeks, F.: Exact polynomial eigenmodes for homogeneous spherical 3-manifolds. Class. Quantum Gravity 23, 6971 (2006) · Zbl 1133.83419 · doi:10.1088/0264-9381/23/23/023 [13] Green, M.B., Schwarz, J., Witten, E.: Superstring Theory. Cambridge University Press, Cambridge (1987) · Zbl 0619.53002 [14] Go, A., Bay, A., et al. (the Belle Collaboration): Measurement of Einstein-Podolsky-Rosen-type flavor entanglement in $$\Upsilon(4S)\rightarrow B^{0}\bar{B}^{0}$$ decays. Phys. Rev. Lett. 99, 131802 (2007) · doi:10.1103/PhysRevLett.99.131802
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