Classification of four qubit states and their stabilisers under SLOCC operations

TL;DR

This paper provides a comprehensive classification of four-qubit states under SLOCC operations, identifying 87 distinct classes with detailed stabiliser information, advancing understanding of quantum entanglement structures.

Contribution

It introduces a complete, irredundant classification of four-qubit states under SLOCC, using symmetric space methods, and details the stabilisers and orbit structures.

Findings

87 distinct classes of four-qubit states identified

Complete classification of stabilisers for each class

Method applicable to similar orbit classification problems

Abstract

We classify four qubit states under SLOCC operations, that is, we classify the orbits of the group $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$ 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 $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$-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 $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$. We also present a complete and irredundant classification of elements and stabilisers up to the action of ${\rm Sym}_4\ltimes\mathrm{\mathop{SL}}(2,\mathbb{C})^4$ where ${\rm Sym}_4$ permutes the four tensor factors of $(\mathbb{C}^2)^{\otimes 4}$.