Here, we only show the *full-dimensional* entanglement polytopes, corresponding to genuinely four-qubit entangled quantum states in \(\mathbb C^2 \otimes \mathbb C^2 \otimes \mathbb C^2 \otimes \mathbb C^2\).

No. |
Class |
Description |
Vertices |
Faces |
Perms. |

1 |
\( L_{0_{7\oplus \bar{1}}} \) |
Origin (0.5,0.5,0.5,0.5) replaced by (0.75,0.5,0.5,0.5). |
12 |
13 |
4 |

2 |
\( L_{0_{5\oplus 3}} \) |
Vertices (0.5,0.5,0.5,0.5) and (1,0.5,0.5,0.5) missing. |
10 |
13 |
4 |

3 |
\( L_{a_4} \) (\( a=0 \)) |
Vertices (0.5,1,0.5,0.5), (0.5,0.5,0.5,1), (0.5,0.5,0.5,0.5) missing. |
9 |
14 |
6 |

4 |
\( L_{\text{ab}_3} \) (\( b=-a \)) |
All three-partite entangled vertices missing. |
8 |
16 |
1 |

5 |
\( L_{\text{ab}_3} \) (\( a=b=0 \)) |
W-state. All three- and four-partite entangled vertices missing. |
7 |
9 |
1 |

6 |
\( L_{a_2b_2} \) (\( b = -a \)) |
Vertices (0.5,1,0.5,0.5), (0.5,0.5,0.5,1) missing. |
10 |
14 |
6 |

7 |
\( G_{\text{abcd}} \) (generic) |
Generic polytope. |
12 |
12 |
1 |

Finally, explore a fermionic system