Classification of four qubit states and their stabilisers under SLOCC operations

• URL: http://arxiv.org/abs/2111.05488v1
• Date: Wed, 10 Nov 2021 02:20:52 GMT
• Title: Classification of four qubit states and their stabilisers under SLOCC operations
• Authors: Heiko Dietrich, Willem A. de Graaf, Alessio Marrani and Marcos Origlia
• Abstract summary: We classify the orbits of the group $mathrmmathopSL (2,mathbbC)4$ on the Hilbert space $mathcalH_4. We also present a complete and irredundant classification of elements and stabilisers up to the action of $rm Sym_4ltimesmathrmmathopSL (2,mathbbC)4$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $(\mathbb{C}^2)^{\otimes 4}$.

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