The twistor geometry of three-qubit entanglement

Abstract

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 $\mathcal{Q}$. An invariant vanishing for the $GHZ$ class is proposed. A geometric interpretation of the canonical decomposition and the inequality for distributed entanglement is also given.

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