Normal Forms and Tensor Ranks of Pure States of Four Qubits

We examine the SLOCC classification of the (non-normalized) pure states of four qubits obtained by F. Verstraete et al. The rigorous proofs of their basic results are provided and necessary corrections implemented. We use Invariant Theory to solve the problem of equivalence of pure states under SLOCC transformations of determinant 1 and qubit permutations. As a byproduct, we produce a new set of generators for the invariants of the Weyl group of type F_4. We complete the determination of the tensor ranks of 4-qubit pure states initiated by J.-L. Brylinski. As a result we obtain a simple algorithm for computing these ranks. We obtain also a very simple classification of pure states of rank at most 3.