© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Three-qubit entangled embedding of CPT and Dirac groups within E8 Weyl group

Michel R. P. Planat

Published Jun 15, 2009• 0 citations

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 encode the new representation of the Dirac group and its CPT subgroup.

Related papers

Balanced Tripartite Entanglement, the Alternating Group A4 and the Lie Algebra sl(3, C) ⊕ u(1)

Michel R. P. Planat, Péter Lévay, Метод Санига

Abstract. We discuss three important classes of three-qubit entangled states and their encoding into quantum gates, finite groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class B of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographic group W(E8) in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates, Preprint 0904.3691 (quant-ph)]. Then, we describe a peculiar “condensation ” of W(E8) into the four-letter alternating group A4, obtained from a chain of maximal subgroups. Group A4 is realized from two B-type generators and found to correspond to the Lie algebra sl(3, C) ⊕ u(1). Possible applications of our findings to particle physics and the structure of genetic code are also mentioned. 1.

•Jan 1, 2009•0 citations

Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates

Michel R. P. Planat

Peres/Mermin arguments about no-hidden variables in quantum mechanics are used for displaying a pair (R, S) of entangling Clifford quantum gates, acting on two qubits. From them, a natural unitary representation of Coxeter/Weyl groups W(D5) and W(F4) emerges, which is also reflected into the splitting of the n-qubit Clifford group Cn into dipoles C$\pm$n . The union of the three-qubit real Clifford group C+ 3 and the Toffoli gate ensures a orthogonal representation of the Weyl/Coxeter group W(E8), and of its relatives. Other concepts involved are complex reflection groups, BN pairs, unitary group designs and entangled states of the GHZ family.

arXiv:0904.3691•Apr 23, 2009•0 citations

Balanced tripartite entanglement, the alternating group $A_4$\\ and the Lie algebra $sl(3,\mathbb{C})\oplus u(1)$

Michel R. P. Planat

I feature three important classes of three-qubit entangled states and their encoding into quantum gates, discrete groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. I introduce another generic class B of three-qubit states, that are equally balanced over two and three parties. I show how to realize the largest cristallographic group $W(E_8)$ in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, {\it Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates}, Preprint 0904.3691 (quant-ph)]. Then, I describe a peculiar condensation of $W(E_8)$ to the four-letter alternating group $A_4$, obtained from a chain of maximal subgroups. Group $A_4$ is realized from two B-type generators and also corresponds to the Lie algebra $sl(3,\mathbb{C})\oplus u(1)$. Applications are to particle and gravitation physics, and to the genetic code.

•Nov 30, 2009•0 citations

Clifiord Group Dipoles and the Enactment of Weyl/Coxeter Group W(E 8 ) by Entangling Gates

Michel R. P. Planat

Peres/Mermin arguments about no-hidden variables in quantum mechanics are used for displaying a pair (R;S) of entangling Clifiord quantum gates, acting on two qubits. From them, a unitary realization of Coxeter/Weyl groups W(D5) and W(F4) emerges, which is also re∞ected into the splitting of the nqubit Clifiord group Cn into subgroup dipoles C § . Generators of the three-qubit real Clifiord group C + 3 with the Tofioli gate ensures a orthogonal realization of the Weyl/Coxeter group W(E8), and of its relatives. Other concepts involved are complex re∞ection groups, BN pairs, unitary group designs and entangled states of the GHZ, W and chain families.

•Apr 23, 2009•0 citations

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

Michel R. P. Planat

arXiv:0906.1063•International Journal of Theoretical Physics•Feb 23, 2010•12 citations

Classification of the subgroups of the two-qubit Clifford group

Eric Kubischta, Ian Teixeira

We perform a complete classification of all 56 subgroups of the two-qubit Clifford group containing the two-qubit Pauli group. We provide generators for these groups using gates familiar to the quantum information community and we reference these groups against the group libraries provided in GAP. We also list several families of groups in higher levels of the two-qubit Clifford hierarchy.

arXiv:2409.14624•Journal of Mathematical Physics•Dec 1, 2025•2 citations

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 encode the new representation of the Dirac group and its CPT subgroup.

Abstract. We discuss three important classes of three-qubit entangled states and their encoding into quantum gates, finite groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class B of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographic group W(E8) in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates, Preprint 0904.3691 (quant-ph)]. Then, we describe a peculiar “condensation ” of W(E8) into the four-letter alternating group A4, obtained from a chain of maximal subgroups. Group A4 is realized from two B-type generators and found to correspond to the Lie algebra sl(3, C) ⊕ u(1). Possible applications of our findings to particle physics and the structure of genetic code are also mentioned. 1.

Peres/Mermin arguments about no-hidden variables in quantum mechanics are used for displaying a pair (R, S) of entangling Clifford quantum gates, acting on two qubits. From them, a natural unitary representation of Coxeter/Weyl groups W(D5) and W(F4) emerges, which is also reflected into the splitting of the n-qubit Clifford group Cn into dipoles C

\pm

n . The union of the three-qubit real Clifford group C+ 3 and the Toffoli gate ensures a orthogonal representation of the Weyl/Coxeter group W(E8), and of its relatives. Other concepts involved are complex reflection groups, BN pairs, unitary group designs and entangled states of the GHZ family.

Balanced tripartite entanglement, the alternating group

A_4

\\ and the Lie algebra

sl(3,\mathbb{C})\oplus u(1)

I feature three important classes of three-qubit entangled states and their encoding into quantum gates, discrete groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. I introduce another generic class B of three-qubit states, that are equally balanced over two and three parties. I show how to realize the largest cristallographic group

W(E_8)

in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, {\it Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates}, Preprint 0904.3691 (quant-ph)]. Then, I describe a peculiar condensation of

W(E_8)

to the four-letter alternating group

A_4

, obtained from a chain of maximal subgroups. Group

A_4

is realized from two B-type generators and also corresponds to the Lie algebra

sl(3,\mathbb{C})\oplus u(1)

. Applications are to particle and gravitation physics, and to the genetic code.

Peres/Mermin arguments about no-hidden variables in quantum mechanics are used for displaying a pair (R;S) of entangling Clifiord quantum gates, acting on two qubits. From them, a unitary realization of Coxeter/Weyl groups W(D5) and W(F4) emerges, which is also re∞ected into the splitting of the nqubit Clifiord group Cn into subgroup dipoles C § . Generators of the three-qubit real Clifiord group C + 3 with the Tofioli gate ensures a orthogonal realization of the Weyl/Coxeter group W(E8), and of its relatives. Other concepts involved are complex re∞ection groups, BN pairs, unitary group designs and entangled states of the GHZ, W and chain families.