Computing local invariants of quantum-bit systems

We investigate means to describe the nonlocal properties of quantum systems and to test if two quantum systems are locally equivalent. For this we consider quantum systems that consist of several subsystems, especially multiple quantum bits, i.e., systems consisting of subsystems of dimension 2. We compute invariant polynomials, i.e., polynomial functions of the entries of the density operator that are invariant under local unitary operations. As an example, we consider a system of two quantum bits. We compute the Molien series for the corresponding representation, which gives information about the number of linearly independent invariants. Furthermore, we present a set of polynomials that generate all invariants (at least) up to degree 23. Finally, the use of invariants to check whether two density operators are locally equivalent is demonstrated.