Classification of four qubit states and their stabilisers under SLOCC operations

We classify four qubit states under SLOCC operations, that is, we classify the orbits of the group SL(2, C) 4 on the Hilbert space H4 = (C2) ⊗4. We approach the classification by realising this representation as a symmetric space of maximal rank. We first describe general methods for classifying the orbits of such a space. We then apply these methods to obtain the orbits in our special case, resulting in a complete and irredundant classification of SL(2, C) 4-orbits on H4. It follows that an element of (C2) ⊗4 is conjugate to an element of precisely 87 classes of elements. Each of these classes either consists of one element or of a parameterised family of elements, and the elements in the same class all have equal stabiliser in SL(2, C) 4. We also present a complete and irredundant classification of elements and stabilisers up to the action of Sym4 SL(2, C) 4 where Sym4 permutes the four tensor factors of (C2) ⊗4.