Effective Rationality for Local Unitary Invariants of Mixed States of Two Qubits

Abstract: We calculate the field of rational local unitary invariants for mixed states of two qubits, by employing methods from algebraic geometry. We prove that this field is rational (i.e. purely transcendental), and that it is generated by nine algebraically independent polynomial invariants. We do so by constructing a relative section, in the sense of invariant theory, whose Weyl group is a finite abelian group. From this construction, we are able to give explicit expressions for the generating invariants in terms of the Bloch matrix representation of mixed states of two qubits. We also prove similar rationality statements for the local unitary invariants of symmetric mixed states of two qubits. Our results apply to both complex-valued and real-valued invariants.