Topology of the three-qubit space of entanglement types

The three-qubit space of entanglement types is the orbit space of the local unitary action on the space of three-qubit pure states and, hence, describes the types of entanglement that a system of three qubits can achieve. We show that this orbit space is homeomorphic to a certain subspace of ${\mathbb{R}}^6$, which we describe completely. We give a topologically based classification of three-qubit entanglement types, and we argue that the nontrivial topology of the three-qubit space of entanglement types forbids the existence of standard states with the convenient properties of two-qubit standard states.

Figure 1

Entanglement types for ${\beta }_4=0$.

Figure 2

Entanglement possibilities for ${\beta }_4=1∕8$. Each point in the interior of the deformed tetrahedron has a circle of entanglement types in the $({\beta }_5,{\beta }_6)$ plane, as shown in Fig. 3. Each point on the boundary of the tetrahedron represents a single entanglement type.

Figure 3

Each point on this deformed circle is an entanglement type with ${\beta }_4=1∕8$, ${\beta }_1={\beta }_2={\beta }_3=0.03$. Every point in the interior of the deformed tetrahedron of Fig. 2 has a circle of entanglement types like this. Points on the boundary of Fig. 2 have a single entanglement type.