language: en

Clifford group dipoles and the enactment of Weyl/Coxeter group W (E 8 ) by entangling gates

Michel Planat
Institut FEMTO-ST, CNRS,
32 Avenue de l'Observatoire, F-25044 Besançon, France
(planat@femto-st.fr)

Abstract. 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 (D 5 ) and W (F 4 ) emerges, which is also reflected into the splitting of the n-qubit Clifford group C n into dipoles C � 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 (E 8 ), and of its relatives. Other concepts involved are complex reflection groups, BN pairs, unitary group designs and entangled states of the GHZ, W and chain families.

PACS numbers: 03.67.Pp, 03.67.Pp, 02.20.-a, 03.65.Ud

1. Introduction