Package of facts and theorems for efficiently generating entanglement criteria for many qubits

We present a package of mathematical theorems that allows us to construct multipartite-entanglement criteria. Importantly, establishing bounds for certain classes of entanglement does not take an optimization over continuous sets of states. These bonds are found from the properties of commutativity graphs of operators used in the criterion. We present two examples of criteria constructed according to our method. One of them detects genuine five-qubit entanglement without ever referring to correlations between all five qubits.

Figure 1

The $\left(A\right|BCDE)$-commutation diagram for operators from set (10). Due to the cyclic permutation symmetry of $\sigma$, analogous graphs apply to all other $\left(4\right|1)$ cuts.

Figure 2

The $\left(AB\right|CDE)$-commutation diagram for operators from set (10). Due to the cyclic permutation symmetry of $\sigma$, analogous graphs apply to all other $\left(3\right|2)$ cuts with two neighboring qubits at one side.

Figure 3

The $\left(AC\right|BDE)$-commutation diagram for operators from set (10). Due to the cyclic permutation symmetry of $\sigma$, similar graphs apply to all other $\left(3\right|2)$ cuts with two non-neighboring qubits at one side.

Figure 4

The commutation diagram for operators from set (10).