Table 1 The number of basis elements of \(\mathcal {H} = \mathcal {H}_2 \otimes \mathcal {H}_2 \otimes \mathcal {H}_2\) that are included in the linear superposition (1) gives rise to a classification of three-qubit states7,8. Some of the representative states generated by such basis elements are associated with specific convex subspaces of the entanglement–polytope \(\mathcal {P} \subset \mathbb {R}^3\) shown in Figure 1, see27 and discussion in the main text. The class of fully separable states is represented by \(\vert 000 \rangle\) as a generic case, see details in Supplementary Information file. States of type 4d are represented by the vector \(\vert \psi _{4d} \rangle\) defined in (5).
From: A geometric formulation to measure global and genuine entanglement in three-qubit systems
Type | Basis product states | Subset \(\subseteq \mathcal {P}\) |
|---|---|---|
1 | \(\lbrace \vert 000 \rangle \rbrace\) | \(\vec {S}\) |
2a-1 | \(\lbrace \vert 101 \rangle , \vert 110 \rangle \rbrace\) | \(\overline{SB_{1}}\) |
2a-2 | \(\lbrace \vert 000 \rangle , \vert 101 \rangle \rbrace\) | \(\overline{SB_{2}}\) |
2a-3 | \(\lbrace \vert 000 \rangle , \vert 110 \rangle \rbrace\) | \(\overline{SB_{3}}\) |
2b | \(\lbrace \vert 000 \rangle , \vert 111 \rangle \rbrace\) | \(\overline{SG}\) |
3a | \(\lbrace \vert 000 \rangle , \vert 101 \rangle , \vert 110 \rangle \rbrace\) | Facets of \(SB_{1}B_{2}B_{3}\) |
3b-1 | \(\lbrace \vert 000 \rangle , \vert 100 \rangle , \vert 111 \rangle \rbrace\) | \(SB_{1}G\) |
3b-2 | \(\lbrace \vert 000 \rangle , \vert 101 \rangle , \vert 111 \rangle \rbrace\) | \(SB_{2}G\) |
3b-3 | \(\lbrace \vert 000 \rangle , \vert 110 \rangle , \vert 111 \rangle \rbrace\) | \(SB_{3}G\) |
4a | \(\lbrace \vert 000 \rangle , \vert 100 \rangle , \vert 101 \rangle , \vert 110 \rangle \rbrace\) | \(SB_{1}B_{2}B_{3}\) |
4b-1 | \(\lbrace \vert 000 \rangle , \vert 100 \rangle , \vert 110 \rangle , \vert 111 \rangle \rbrace\) | \(\subset SB_{1}B_{3}G\) |
4b-2 | \(\lbrace \vert 000 \rangle , \vert 100 \rangle , \vert 101 \rangle , \vert 111 \rangle \rbrace\) | \(\subset SB_{1}B_{2}G\) |
4c | \(\lbrace \vert 000 \rangle , \vert 101 \rangle , \vert 110 \rangle , \vert 111 \rangle \rbrace\) | \(SB_{1}B_{2}B_{3} \cup SB_{2}B_{3}G\) |
4d | \(\lbrace \vert 001 \rangle , \vert 010 \rangle , \vert 100 \rangle , \vert 111 \rangle \rbrace\) | \(\mathcal {P}\) |
5 | \(\lbrace \vert 000 \rangle , \vert 100 \rangle , \vert 101 \rangle , \vert 110 \rangle ,\vert 111 \rangle \rbrace\) | \(\subset \mathcal {P}\) |
