Rational Local Unitary Real Invariants of Mixed States of Two Qubits

We compute the field of rational local unitary invariants for several types of mixed states of two qubits that are of interest in quantum information theory. In particular, we consider locally maximally mixed states, symmetrically mixed states and the full space of mixed states. In each case, we construct explicit geometric quotients and prove that they are always rational. As a consequence, we prove in each case that the field of rational invariants is purely transcendental. All the results are obtained by working over the field of real numbers, employing methods from classical and geometric invariant theory over arbitrary fields of characteristic zero.