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

The Shor-Laflamme distribution (SLD) of a quantum state is a collection of local unitary invariants that quantify k-body correlations. We show that the SLD of graph states can be derived by solving a graph-theoretical problem. In this way, the mean and variance of the SLD are obtained as simple functions of efficiently computable graph properties. Furthermore, this formulation enables us to derive closed expressions of SLDs for some graph state families. For cluster states, we observe that the SLD is very similar to a binomial distribution, and we argue that this property is typical for graph states in general. Finally, we derive an SLD-based entanglement criterion from the purity criterion and apply it to derive meaningful noise thresholds for entanglement. Our new entanglement criterion is easy to use and also applies to the case of higher-dimensional qudits. In the bigger picture, our results foster the understanding both of quantum error-correcting codes, where a closely related notion of SLDs plays an important role, and of the geometry of quantum states, where SLDs are known as sector length distributions.

Automated summary

The Shor-Laflamme distribution is a set of local unitary invariants that measure k-body correlations in a quantum state. We demonstrate that the distribution of graph states can be obtained by solving a graph-theoretical problem, allowing for the calculation of mean and variance using graph properties. We also derive closed expressions for the distribution of certain graph state families. Our results provide insights into quantum error-correcting codes and the geometry of quantum states. We propose an entanglement criterion based on the Shor-Laflamme distribution, which can be applied to higher-dimensional systems.