Characterizing quantum networks: Insights from coherence theory

Networks based on entangled quantum systems enable interesting applications in quantum information processing and the understanding of the resulting quantum correlations is essential for advancing the technology. We show that the theory of quantum coherence provides powerful tools for analyzing this problem. For that, we demonstrate that a recently proposed approach to network correlations based on covariance matrices can be improved and analytically evaluated for the most important cases.

Figure 1

(a) The triangle network consists of three nodes that produce measurement outcomes $x_1,\cdots ,x_3$ and three sources that distribute bipartite entanglement that is shared amongst the nodes. The covariance matrix of the triangle network has a $3\times 3$ block structure and consists of three terms, where ${\left(\square \right)}_i$ denotes those blocks that are contributed by the source $i$. (b) A network consisting of four nodes that is 3-complete, i.e., it features four sources that distribute tripartite entanglement.

Figure 2

(a) For the triangle network we compare the criterion in Eq. (7) (solid line) to the monogamy criterion in Eq. (10) (dashed line), the entropic constraint (dot-dashed line) from Refs. [32, 33], and the inflation criterion (dotted line) from Ref. [10]. (b) Results of the GHZ-type distribution in Eq. (5) using Eq. (8) for $N=5$ and $k=4,3,2$. Everything above the lines is detected to be incompatible with the respective network structure.

Figure 3

Analysis of the GHZ-type distribution with three outcomes per measurement in Eq. (9) using Observation t4. The blue surface represents the normalization constraint.