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 $\mathrm{SL}(2,\mathbb{C}{)}^4$ on the Hilbert space ${\mathcal{H}}_4=({\mathbb{C}}^2{)}^{\otimes 4}$. 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 $\mathrm{SL}(2,\mathbb{C}{)}^4$-orbits on ${\mathcal{H}}_4$. It follows that an element of $({\mathbb{C}}^2{)}^{\otimes 4}$ is conjugate to an element of precisely 87 classes of elements. Each of these classes either consists of one element or of a parametrised family of elements, and the elements in the same class all have equal stabiliser in $\mathrm{SL}(2,\mathbb{C}{)}^4$. We also present a complete and irredundant classification of elements and stabilisers up to the action of ${\mathrm{Sym}}_4⋉\mathrm{SL}(2,\mathbb{C}{)}^4$ where ${\mathrm{Sym}}_4$ permutes the four tensor factors of $({\mathbb{C}}^2{)}^{\otimes 4}$.

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Cited by in corpus (2)

• Tensor decompositions with applications to LU and SLOCC equivalence of multipartite pure states
• Toward Jordan Decompositions of Tensors