  In this paper we study universality for quantum gates acting on qudits.Qudits
are states in a Hilbert space of dimension d where d is at least two. We
determine which 2-qudit gates V have the properties (i) the collection of all
1-qudit gates together with V produces all n-qudit gates up to arbitrary
precision, or (ii) the collection of all 1-qudit gates together with V produces
all n-qudit gates exactly. We show that (i) and (ii) are equivalent conditions
on V, and they hold if and only if V is not a primitive gate. Here we say V is
primitive if it transforms any decomposable tensor into a decomposable tensor.
We discuss some applications and also relations with work of other authors.
