Geometric picture for SLOCC classification of pure permutation symmetric three-qubit states

TL;DR

This paper introduces a geometric approach using spheroids within the Bloch sphere to classify pure permutation symmetric three-qubit states under SLOCC, revealing distinct geometric signatures for different state classes.

Contribution

It provides a novel geometric representation for SLOCC classes of symmetric three-qubit states based on Lorentz canonical forms and spheroids inside the Bloch sphere.

Findings

Distinct spheroid geometries correspond to different SLOCC classes.

Pure states from 3 distinct spinors form a prolate spheroid centered at the origin.

States from 2 distinct spinors form a spheroid centered at (0,0,1/2) with fixed axes.

Abstract

We show that the pure entangled three-qubit symmetric states which are inequivalent under stochastic local operations and classcial communication (SLOCC) exhibit distinct geometric representation in terms of a spheroid inscribed within the Bloch sphere. We provide detailed analysis of the SLOCC canonical forms of the reduced two-qubit states extracted from entangled three-qubit pure symmetric states. Based on the Lorentz canonical forms of these states we arrive at two different geometrical representations: (i) a prolate spheroid centered at the origin of the Bloch sphere -- with longest semiaxis along the z-direction (symmetry axis of the spheroid) equal to 1 -- in the case of pure permutation symmetric three-qubit states constructed from 3 distinct spinors and (ii) a spheroid centered at (0,0,1/2) inside the Bloch sphere, with fixed semiaxes lengths (1/sqrt{2}, 1/sqrt{2}, 1/2) when the three-qubit pure state is constructed via symmetrization of 2 distinct spinors.