Classification of four qubit states and their stabilisers under SLOCC operations - Dietrich, Heiko (2022)
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 .