Coxeter groups A4, B4 and D4 for two-qubit systems

Abstract

The Coxeter–Weyl groups W(A4), W(B4), and W(D4) have proven very useful for two-qubit systems in quantum information theory. A simple technique is employed to construct the unitary matrix representations of the groups, based on quaternionic transformation of the usual reflection matrices. The von Neumann entropy of each reduced density matrix is calculated. It is shown that these unitary matrix representations are naturally related to various universal quantum gates and they lead to entangled states. Canonical decomposition of generators in terms of fundamental gate representations is given to construct the quantum circuits.