Tensor Rank and Other Multipartite Entanglement Measures of Graph States

TL;DR

This paper investigates multipartite entanglement in graph states, providing new bounds on tensor rank, characterizing entanglement measures, and deriving a simple rule for computing the n-tangle.

Contribution

It introduces tighter bounds on tensor rank for odd ring states and characterizes multipartite entanglement measures based on graph connectivity.

Findings

Bounds on tensor rank of odd ring states are tightened.

Multipartite entanglement measures are dichotomous based on graph connectivity.

A simple rule for computing the n-tangle is provided.

Abstract

Graph states play an important role in quantum information theory through their connection to measurement-based computing and error correction. Prior work has revealed elegant connections between the graph structure of these states and their multipartite entanglement content. 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 ($|R_{2n+1}\rangle$) to read $2^n+1 \leq rank(|R_{2n+1}\rangle) \leq 3*2^{n-1}$. Next, we show that several multipartite extensions of bipartite entanglement measures are dichotomous for graph states based on the connectivity of the corresponding graph. Lastly, we give a simple graph rule for computing the n-tangle $\tau_n$.