Structure of Clifford groups of composite finite quantum systems

In this paper, the Clifford groups of general composite quantum systems with the configuration space Z n 1 x ... x Z n k , k >=1, are studied. It is known that the Clifford group is a natural semidirect product, provided that the dimension N = n 1 • • • n k of the corresponding Hilbert space is odd. For even N special results on the Clifford groups are scattered in the mathematical literature, but they do not concern the semidirect product structure. Using the relation between the generators of the associated symplectic group Sp [n 1 ,...,n k ] we prove that for even N = n 1 • • • n k both the Clifford group and the projective Clifford group are natural semidirect products if and only if N is not divisible by four.