We improve some results relative to the state complexity of the multiple catenations described by Gao and Yu. In particular we nearly divide by 2 the size of the alphabet needed for witnesses. We also give some refinements to the algebraic expression of the state complexity, which is especially complex with this operation. We obtain these results by using peculiar DFAs defined by Brzozowski.

We give an algorithm allowing the construction of bases of local unitary invariants of pure k -qubit states from a knowledge of the polynomial covariants of the group of invertible local filtering operations. The simplest invariants obtained in this way are made explicit and compared with various known entanglement measures. Complete sets of generators are obtained for up to four qubits, and the structure of the invariant algebras is discussed in detail.