Structure of Multidimensional Entanglement in Multipartite Systems

We explore the structure of multipartite quantum systems which are entangled in multiple degrees of freedom. We find necessary and sufficient conditions for the characterization of tripartite systems and necessary conditions for any number of parties. Furthermore we develop a framework of multilevel witnesses for efficient discrimination and quantification of multidimensional entanglement that is applicable for an arbitrary number of systems and dimensions.

Figure 1

Schematic representation of a few sets of states with a given Schmidt number vector.

Figure 2

Here we depict the noise resistance of our detection method in the presence of white and dephasing noise for the exemplary state from Eq. (10). In the top left corner the state is pure; going down vertically adds white noise and going diagonally to the right dephases the state. The differently colored (shaded) regions are labeled with the corresponding lower bounds on the dimensionalities that one can prove using the nonlinear witness from Eq. (9).