Classification of four qubit states and their stabilisers under SLOCC operations

Abstract

Abstract We classify four qubit states under SLOCC operations, that is, we classify the orbits of the group S L ( 2 , C ) 4 on the Hilbert space H 4 = ( C 2 ) ⊗ 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 S L ( 2 , C ) 4 -orbits on H 4 . It follows that an element of ( C 2 ) ⊗ 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 S L ( 2 , C ) 4 . We also present a complete and irredundant classification of elements and stabilisers up to the action of S y m 4 ⋉ S L ( 2 , C ) 4 where Sym4 permutes the four tensor factors of ( C 2 ) ⊗ 4 .