Freudenthal triple classification of three-qubit entanglement

Abstract

We show that the three-qubit entanglement classes, (0) null, (1) separable $A\text{\ensuremath{-}}B\text{\ensuremath{-}}C$, (2a) biseparable $A\text{\ensuremath{-}}BC$, (2b) biseparable $B\text{\ensuremath{-}}CA$, (2c) biseparable $C\text{\ensuremath{-}}AB$, (3) $W$, and (4) Greenberger-Horne-Zeilinger, correspond respectively to ranks 0, 1, 2a, 2b, 2c, 3, and 4 of a Freudenthal triple system defined over the Jordan algebra $\mathbb{C}\ensuremath{\bigoplus}\mathbb{C}\ensuremath{\bigoplus}\mathbb{C}$. We also compute the corresponding stochastic local operations and classical communication orbits.