Clifford Group Dipoles and the Enactment of Weyl/Coxeter Group W(E8) by Entangling Gates

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 unitary realization 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 subgroup dipoles C^+-n . Generators of the three-qubit real Clifford group C^ 3 with the Toffoli gate ensures a orthogonal realization of the Weyl/Coxeter group (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.