Yokohama, 236–0027

In this paper we revisit the isomorphism SU(2) ⊗ SU(2) ∼= SO(4) to apply to some subjects in Quantum Computation and Mathematical Physics. The unitary matrix Q by Makhlin giving the isomorphism as an adjoint action is studied and generalized from a different point of view. Some problems are also presented. In particular, the homogeneous manifold SU(2n)/SO(2n) which characterizes entanglements in the case of n = 2 is studied, and a clear–cut calculation of the universal E-mail address : [email protected] E-mail address : [email protected] E-mail address : [email protected] 1 Yang–Mills action in (hep-th/0602204) is given for the abelian case.