Quantifying Genuine Multipartite Correlations and their Pattern Complexity

We propose an information-theoretic framework to quantify multipartite correlations in classical and quantum systems, answering questions such as what is the amount of seven-partite correlations in a given state of ten particles? We identify measures of genuine multipartite correlations, i.e., statistical dependencies that cannot be ascribed to bipartite correlations, satisfying a set of desirable properties. Inspired by ideas developed in complexity science, we then introduce the concept of weaving to classify states that display different correlation patterns, but cannot be distinguished by correlation measures. The weaving of a state is defined as the weighted sum of correlations of every order. Weaving measures are good descriptors of the complexity of correlation structures in multipartite systems.

Figure 1

Multipartite correlation hierarchy. Given a system of $N$ particles (blue spheres), the sets $P_k$, $k=1,2,\dots ,N$, consist of states displaying up to $k$-partite correlations. The yellow $k$ spheres identify the largest subset of a coarse grained partition (the dashed red lines separate each cluster). The amount of genuine $k$-partite correlations in a state is the difference between the distance to the sets $P_{k-1}$ and $P_k$.