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The Shor-Laflamme distribution (SLD) of a quantum state is a collection oflocal unitary invariants that quantify $k$-body correlations. We show that theSLD of graph states can be derived by solving a graph-theoretical problem. Inthis way, the mean and variance of the SLD are obtained as simple functions ofefficiently computable graph properties. Furthermore, this formulation enablesus to derive closed expressions of SLDs for some graph state families. Forcluster states, we observe that the SLD is very similar to a binomialdistribution, and we argue that this property is typical for graph states ingeneral. Finally, we derive an SLD-based entanglement criterion from the puritycriterion and apply it to derive meaningful noise thresholds for entanglement.Our new entanglement criterion is easy to use and also applies to the case ofhigher-dimensional qudits. In the bigger picture, our results foster theunderstanding both of quantum error-correcting codes, where a closely relatednotion of Shor-Laflamme distributions plays an important role, and of thegeometry of quantum states, where Shor-Laflamme distributions are known assector length distributions.

Shor-Laflamme distributions of graph states and noise robustness of entanglement