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 P 2 , respectively. It has been found (Socolovsky, Int. J. Theor. Phys. 43: 1941 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 (E 8 ) is generated (Planat, Preprint 0904.3691, 2009). In this paper, one derives three-qubit entangling groupsP and P 2 , isomorphic to the CPT group P and to the Dirac group 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.

Balanced tripartite entanglement, the alternating group A4 and the lie algebra sl(3,ℂ)⊕u(1)

“…5) is put forward. Earlier papers of one of the authors focused on the three-qubit realization of Coxeter groups, such as the largest one W (E 8 ), together with its most relevant subgroups comprising the three-qubit Clifford group C + 3 , W (E 7 ), W (E 6 ), W (F 4 ) and other subgroups [7,9]. Here, one discovers that the twoqubit real entangling gate S 2 [see Eq.…”

“…As described in papers [7,9], these representations find their kernel in two-qubit entanglement and the following orthogonal matrix…”

“…In this paper, we are especially interested in a class of entangled three-qubit states displaying equally distributed entanglement about three and two parties. Such states were already encountered in the context of CPT symmetry [9]. Here, they occur when one "condenses" the three-qubit representation of W (E 8 ) to the alternating group A 4 , through an appropriate chain of maximal subgroups.…”

“…§ The second largest subgroup of W (E 8 ) is the real Clifford group C + 3 , of order 2 580 480 studied in [7,9].…”

Kolmogorov space in time series data

“…Many scientists from signal processing [28] start to use quaternions for time series data analysis. In physics, there exists a tensor field on Hilbert space for processing information, the so-called qubit states of quantum information [29].…”