Tensor Rank and Other Multipartite Entanglement Measures of Graph States

TL;DR

This work advances the understanding of multipartite entanglement in graph states by (i) sharpening the CP tensor-rank bounds for odd-ring graph states from the prior $2^{n} \leq \mathrm{rank}(|R_{2n+1}\rangle) \leq 2^{n+1}$ to $2^{n}+1 \leq \mathrm{rank}(|R_{2n+1}\rangle) \leq 3\cdot 2^{n-1}$, using a novel combination of line-state decompositions and subsystem-rank analysis; and (ii) showing that multipartite extensions of bipartite entanglement measures (GME concurrence, GME negativity, GME geometric measure) are dichotomous for graph states, yielding 0 for disconnected graphs or fixed constants for connected graphs, along with a simple, graph-rule-based computation of the n-tangle $\tau_n$. The results provide both tighter quantitative characterizations of graph-state entanglement and a tractable path to evaluating these measures for stabilizer states, with implications for LOCC entanglement cost and measurement-based quantum computing. The paper also clarifies an efficient stabilizer-based method to compute $\tau_n$ and connects graph topology (connectivity and degree parity) to entanglement content in a precise, operational way.

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 $τ_n$.

Figures (5)

• Figure 1: Example of local complementation. Here the rule is applied to the red vertex, adding or removing the edge connecting the two other vertices.
• Figure 2: Line and ring graphs. The left graph is a line (one dimensional cluster state) on 7 qubits, which we denote by $\ket{L_{7}}$. The right graph is an odd ring on 5 qubits, which we denote by $\ket{R_{5}}$.
• Figure 3: Lower bounding the rank of $\ket{R_{2n+1}}$ for $n=3$. For the seven-qubit ring state, we trace out the subset of qubits $A=\{2,4,6\}$, depicted as the red nodes in the ring. This set is judiciously chosen such that after tracing out qubits belonging to $A$, the remaining qubits $\overline{A}=\{1,3,5,7\}$ (shown in blue) consists of just one entangled pair $(1,7)$ and the rest are completely uncoupled. This relatively simple structure allows us to characterize all the product states in the support of $\rho^{(\overline{A})}$ (Lemma \ref{['Lem:product-states-support']}). Using this characterization, the desired lower bound on the tensor rank is proven (Theorem \ref{['Thm:rank-lb']}).
• Figure 4: Graphs with $\tau_n=1$. All vertices in the graphs above have odd degree and thus have maximal n-tangle via Thm. \ref{['thm:n_tangle']}.
• Figure 5: GHZ state graph. For any system size, the GHZ state is local Clifford equivalent to a star graph state. Through the local complementation rule, this is also local Clifford equivalent to the fully connected graph.

Theorems & Definitions (34)

• lemma 1.1: hein2004multiparty
• lemma 1.2
• proof
• theorem 2.1
• Proposition 2.2: CP rank upper bound for odd ring graph states
• proof
• lemma 2.3
• proof
• lemma 2.4
• proof