# Introduction

This is a database of entanglement in graph states of up to 12 qubits, the result of a joint work by A. Cabello, L.E. Danielsen, A.J. López-Tarrida, and J.R. Portillo. For more details, see the paper [4]. Entanglement in graph states has previously been classified for states of up to 7 qubits [1], and for 8 qubits [2].

We also list cardinality-multiplicity invariants that distinguish graph states. (Note that for 9 or more qubits, there is a small number of pairs of exceptional states that are not distinguished [4].) Cardinality-multiplicity invariants were first defined and calculated for up to 8 qubits in [3].

^ TOP# File format

The file format is as follows. See [4] for more detailed definitions. Each line gives data about one graph state, and has the following format:

- No.: Number of the equivalence class.
- |LC|: Number of nonisomorphic graphs in the class.
- |V|: Number of vertices.
- min(|E|, χ', #): |E| is the minimum number of edges in the class. χ' is the minimum chromatic index of the graphs with |E| edges. # is the number of nonisomorphic graphs with |E| edges and chromatic index χ'.
- min(χ', |E|, #): χ' is the minimum chromatic index in the class. |E| is the minimum number of edges of the graphs with chromatic index χ'. # is the number of nonisomorphic graphs with chromatic index χ' and |E| edges. (This field is left blank if the values are identical to the previous field.)
- E
_{S}: Schmidt measure. (Upper and lower bound is given where the exact value is unknown.) - RI
_{i}: (for n/2 ≥ i ≥ 2): Rank index for bipartite splits with i vertices in the smaller partition. - C-M: (for 0 ≤ i ≤ x) Cardinality-multiplicities. Value i is the multiplicity of the cardinality i. Only the multiplicities of cardinalities 0 to x are listed, where x is the smallest number such that all graph states that can be distinguished are distinguished.
- 2-col: Does the class contain a two-colorable graph?
- A representive graph from the class with minimum number of edges. The edges of the graph are listed (with vertices labeled 0 to n-1). Edges are collected in classes (enclosed by parentheses) that define a proper edge-coloring with the minimum number of colors.
- A representive graph from the class with minimum chromatic index. (This field is left blank if the graph in the previous field has minimum chromatic index.)

# Files

n | Download | Size |
---|---|---|

2 | entanglement2 | 1 graph |

3 | entanglement3 | 1 graph |

4 | entanglement4 | 2 graphs |

5 | entanglement5 | 4 graphs |

6 | entanglement6 | 11 graphs |

7 | entanglement7 | 26 graphs |

8 | entanglement8 | 101 graphs |

9 | entanglement9 | 440 graphs |

10 | entanglement10 | 3132 graphs (509 KB) |

11 | entanglement11.bz2 | 40,457 graphs (1.2 MB compressed) |

12 | entanglement12.bz2 | 1,274,068 graphs (45 MB compressed) |

We have also written a computer program that, given a graph, finds optimal representatives in the LC orbit (in terms of number of edges and chromatic index) and gives LC-sequence(s) producing the input graph from the optimal graph(s):

Download | |
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findoptimal.c | C source code |

# References

[1] M. Hein, J. Eisert, and H.J. Briegel. "Multiparty entanglement in graph states." Phys. Rev. A 69, 062311 (2004)

[2] A. Cabello, A.J. López-Tarrida, P. Moreno, and J.R. Portillo. "Entanglement in eight-qubit graph states." Phys. Lett. A 373, 2219-2225 (2009)

[3] A. Cabello, A.J. López-Tarrida, P. Moreno, and J.R. Portillo. "Compact set of invariants characterizing graph states of up to eight qubits." Phys. Rev. A 80, 012102 (2009)

[4] Adán Cabello, Lars Eirik Danielsen, Antonio J. López-Tarrida, and José R. Portillo. Optimal preparation of graph states. Nov. 2010. (Submitted for publication) (arXiv)