Classification of four-rebit states

TL;DR

This paper classifies the orbits of four-rebit states under a real SLOCC group, dividing them into semisimple, nilpotent, and mixed types, with implications for black hole solutions in supergravity.

Contribution

It provides a new classification of semisimple and mixed orbits of four-rebit states using Galois cohomology, extending previous nilpotent orbit classifications.

Findings

Classification of nilpotent orbits completed in prior work.

Semisimple and mixed orbits classified with Galois cohomology methods.

Results applicable to black hole solution types in supergravity.

Abstract

We classify states of four rebits, that is, we classify the orbits of the group $\widehat{G}(\mathbb R) = \mathrm{\mathop{SL}}(2,\mathbb R)^4$ in the space $(\mathbb R^2)^{\otimes 4}$. This is the real analogon of the well-known SLOCC operations in quantum information theory. By constructing the $\widehat{G}(\mathbb R)$-module $(\mathbb R^2)^{\otimes 4}$ via a $\mathbb Z/2\mathbb Z$-grading of the simple split real Lie algebra of type $D_4$, the orbits are divided into three groups: semisimple, nilpotent and mixed. The nilpotent orbits have been classified in Dietrich et al. (2017), yielding applications in theoretical physics (extremal black holes in the STU model of $\mathcal{N}=2, D=4$ supergravity, see Ruggeri and Trigiante (2017)). Here we focus on the semisimple and mixed orbits which we classify with recently developed methods based on Galois cohomology, see Borovoi et al. (2021). These orbits are relevant to the classification of non-extremal (or extremal over-rotating) and two-center extremal black hole solutions in the STU model.