Tensor rank and other multipartite entanglement measures of graph states

Graph states play an important role in quantum information theory through their connection to measurement-based computing and error correction. Prior work revealed elegant connections between the graph structure of these states and their multipartite entanglement. We continue this line of investigation by identifying additional entanglement properties for certain types of graph states. From the perspective of tensor theory, we tighten both upper and lower bounds on the tensor rank of odd ring states ($|{R}_{2n+1}\ensuremath{\rangle}$) to read ${2}^{n}+1\ensuremath{\le}\mathrm{rank}(|{R}_{2n+1}\ensuremath{\rangle})\ensuremath{\le}3\ifmmode\times\else\texttimes\fi{}{2}^{n\ensuremath{-}1}$. Next we show that several multipartite extensions of bipartite entanglement measures are dichotomous for graph states based on the connectivity of the corresponding graph. Finally, we give a simple graph rule for computing the $n$-tangle ${\ensuremath{\tau}}_{n}$.