Miroslav Korbelář
Structure of Clifford groups of composite finite quantum systems
(a joint work with J. Tolar)
We consider the Clifford groups of general multipartite quantum systems with configuration space $\mathbb{Z}_{n_1}\oplus\cdots\oplus\mathbb{Z}_{n_k}$. It is known that the Clifford group is a natural semidirect product provided the dimension $N=n_1\cdots n_k$ of the corresponding Hilbert space is an odd number.
For even $N$ special results on the Clifford groups are scattered
in the mathematical literature, but they do not concern the semidirect product structure. We show that for even $N=n_1\cdots n_k$ both Clifford group and the projective Clifford group are natural semidirect products if and only if $N$ is not divisible by four. Our approach is based on Based on relation of generators of the associated symplectic group $\operatorname{Sp}_{[n_1,\dots,n_k]}$.
