Multipartite entanglement configurations: Combinatorial offshoots into (hyper)graph theory and their ramifications

Computations in a distributed environment comprising a network of spatially separated nodes may require the exchange of classical and quantum information. The amount of classical communication may be reduced in such computations by using multipartite entanglement. Following the combinatorial approach developed in [25, 27], we study entanglement configurations over a set of nodes, where each entanglement configuration is a collection of multipartite entanglement (CAT or GHZ) states shared within different combinations of subsets of nodes. The main problem is to determine whether LOCC transformations can generate an entanglement configuration B from another entanglement configuration A, written as B LOCC A. We characterize the resulting partial order introduced on unitarily equivalent classes of entanglement configurations due to LOCC transformations. This study includes the communication complexity of generating higher cardinality multipartite CAT states from smaller sized CAT state configurations. We also study classes of incomparable entanglement configurations where no pair A B of configurations satisfies A LOCC B. This leads us to investigate certain combinatorial properties of hypergraphs and hypertrees following initial results in [25, 27]. We study the unique reconstruction of vertex labelled r-uniform hypertrees on n vertices, where r n is a constant, and each hyperedge has the same number r, of vertices. We conclude by discussing several problems and open questions in the context of entanglement configurations.