Polynomial-time Extraction of Entanglement Resources

TL;DR

This work addresses the challenge of extracting remote entanglement resources in graph-state quantum networks by formulating the Remote-VM problem and proving its polynomial-time solvability. It introduces two sufficient conditions for remote $n$-Gability and a FindA-driven algorithm that computes the exact extractable volumes of remote $n$-qubit GHZ states and EPR pairs on general two-colorable graphs, with complexity $O(|P_1|^3|P_2|)$. The approach yields flexible extraction across a range of GHZ masses (up to $n$ in $[3,17]$ in experiments) and demonstrates scalability on Internet-inspired topologies, enabling on-demand, dynamic entanglement provisioning via LOCC. The results have practical implications for adaptive quantum networking, where remote entanglement resources must be allocated in real time to satisfy traffic demands.

Abstract

The extraction of EPR pairs and n-qubits GHZ states among remote nodes in quantum networks constitutes the resource primitives for end-to-end and on-demand communications. However, the Bell-VM problem, which determines whether a given graph state can be transformed into a set of Bell pairs on specific vertices (not necessarily remote), is known to be NP-complete. In this paper, we extend this problem, not only by focusing on nodes remote within generic graph states, but also by determining the number of extractable n-qubit remote GHZ states -- beside the number of remote EPR pairs. The rationale for tackling the extraction of GHZ states among remote nodes, rather than solely remote EPR pairs, is that a GHZ state enables the dynamic extraction of an EPR pair between any pair of nodes sharing the state. This, in turn, implies the ability of accommodating the traffic requests on-the-fly. Specially, we propose a polynomial-time algorithm for solving the aforementioned NP-complete problem. Our results demonstrate that the proposed algorithm is able to effectively adapt to generic graph states for extracting entanglement resources across remote nodes.

Figures (15)

• Figure 1: Remote vs vanilla Pairability and Gability for a 5-qubit linear graph state. Fig. \ref{['fig:01.a']} shows the initial artificial topology enabled by a 5-qubits linear graph state. When it comes to Pairability, up to $2$ EPR pairs can be extracted from the considered graph state. Yet, the extracted EPRs in Fig. \ref{['fig:01.b']} "link" nodes that are already connected in the initial artificial topology. Conversely, only one remote EPR in Fig. \ref{['fig:01.c']} can be extracted in the same artificial topology. When it comes to $n$-Gability, the differences are even more crucial. Specifically, GHZ state with mass up to $4$ qubits as shown in Fig. \ref{['fig:01.d']} can be extracted among the nodes, regardless of their proximity in the artificial topology. Conversely, if we constraint the nodes to be remote as in Def. \ref{['def:02']}, the largest GHZ state has mass equal to $3$, as shown in Fig. \ref{['fig:01.e']}.
• Figure 2: Venn diagram for the relationship between GHZ-VM, BELL-VM, and REMOTE-VM (our research problem).
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Theorems & Definitions (14)

• Definition 1: Remote Nodes
• Definition 2: $\bm{{r_g}(n)}$: remote $\bf{n}$-Gability
• Remark
• Definition 3: $\bm{{r_g}(2)}$: remote Pairability
• Definition 4: Two-colorable Graph or Bipartite Graph
• Definition 5: Opposite Remote-Set
• Definition 6: Star vertex
• Theorem 1
• Remark
• Lemma 1: Remote $n$-Gability: Condition I