Entanglement witnesses for stabilizer states and subspaces beyond qubits

Abstract

Genuine multipartite entanglement is arguably the most valuable form of entanglement in the multipartite case, with applications, for instance, in quantum metrology. In order to detect that form of entanglement in multipartite quantum states, one typically uses entanglement witnesses. The aim of this paper is to generalize the results of [G. Tóth and O. Gühne, Phys. Rev. A \textbf{72}, 022340 (2005)] in order to provide a construction of witnesses of genuine multipartite entanglement tailored to entangled subspaces originating from the \textit{multi-qudit} stabilizer formalism -- a framework well known for its role in quantum error correction, which also provides a very convenient description of a broad class of entangled multipartite states (both pure and mixed). Our construction includes graph states of arbitrary local dimension. We then show that in certain situations, the obtained witnesses detecting genuine multipartite entanglement in quantum systems of higher local dimension are superior in terms of noise robustness to those derived for multiqubit states.

Figures (3)

• Figure 1: Exemplary graphs leading to four-partite graph states. Graph (a) represents the GHZ state, (b) the cluster state, and (c) is an example of a graph with chromatic number $K=3$.
• Figure 2: Comparison between the robustness to white noise of the GHZ and the cluster state. The plots above show how $p_{\textrm{limit}}$ of EW for the GHZ state and the cluster state changes with $N$ for four different values of $d$. It can be observed that in each case, the EW for the GHZ state outperforms the EW for the cluster state. Moreover, for large enough $N$, both states asymptotically reach $p_{sat}$, different for each EW. Comparing all four plots, we see that, in both cases, $p_{sat}$ increases with $d$. For large $d$, $p_{sat}$ of both EW will approach $p_{sat} = 1/2$.
• Figure 3: Comparison between the robustness to white noise of the GHZ and the stabilizer subspace (\ref{['eq:gens_opt']}). The four plots above show how $p_{\textrm{limit}}$ changes with $N$ for different $d$ in the case of EWs for the GHZ state and the stabilizer subspace (\ref{['eq:gens_opt']}). It can be observed that for each $d$ the EW for the subspace performs better, but the results approach the same value $p_{sat}$ for large $N$. Moreover, $p_{sat}$ grows with $d$ and it will approach $p_{sat} = 1/2$ for large $d$.

Theorems & Definitions (16)

• Theorem 1
• proof
• Theorem 2
• Lemma 1
• proof
• proof
• Theorem 3
• proof
• Theorem 4
• proof