Bell Inequalities for Graph States

We investigate the nonlocal properties of graph states. To this aim, we derive a family of Bell inequalities which require three measurement settings for each party and are maximally violated by graph states. In turn, for each graph state there is an inequality maximally violated only by that state. We show that for certain types of graph states the violation of these inequalities increases exponentially with the number of qubits. We also discuss connections to other entanglement properties such as the positivity of the partial transpose or the geometric measure of entanglement.

Figure 1

Types of graphs for the case of five vertices: (a) The linear cluster graph $L{C}_5$. (b) The ring cluster graph $R{C}_5$. (c) The star (or GHZ) graph $S{T}_5$. This describes a GHZ state. (d) The fully connected graph $F{C}_5$. This graph can be obtained from $S{T}_5$ by local complementation on the second qubit (see Lemma 4). It also describes a GHZ state [3].