Classification of four qubit states and their stabilisers under SLOCC operations

Abstract

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 .