Geometry of three-qubit entanglement

A geometrical description of three-qubit entanglement is given. A part of the transformations corresponding to stochastic local operations and classical communication on the qubits is regarded as a gauge degree of freedom. Entangled states can be represented by the points of the Klein quadric $\mathcal{Q}$, a space known from twistor theory. It is shown that three-qubit invariants are vanishing on special subspaces of $\mathcal{Q}$ with interesting geometric properties. An invariant characterizing the Greenberger-Horne-Zeilinger class is proposed. A geometric interpretation of the canonical decomposition and the inequality for distributed entanglement is also given.