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On the Efficient Extraction of Entangled Resources

Si-Yi Chen, Angela Sara Cacciapuoti, Marcello Caleffi

TL;DR

The paper tackles the problem of remotely extracting entangled resources (GHZ states and EPR pairs) from a given two-colorable graph state, where remote nodes are non-adjacent in the artificial topology. It formalizes remote $n$-Gability and remote Pairability, establishes NP-completeness, and provides constructive lower and upper bounds on extractable volumes, including the maximum mass of remote GHZ states. A polynomial-time heuristic, Algorithm Remote Extraction, leverages single-qubit Clifford operations, Pauli measurements, and classical communication to compute volumes, identify node locations, and estimate the maximum GHZ size, with proven polynomial-time complexity. Performance evaluations on bipartite and Internet-inspired graphs demonstrate the algorithm yields nontrivial remote resources (e.g., remote GHZ masses from 3 to 17 and multiple EPR pairs) and scales with graph density. The work lays a foundation for dynamic, end-to-end quantum communications by enabling on-demand remote entanglement extraction, and it discusses future extensions to more general graph classes and noisy environments to enhance practicality in real quantum networks.

Abstract

In the Quantum Internet, multipartite entanglement enables a rich and dynamic overlay topology, referred to as artificial topology, upon the physical one, that can be exploited for communication purposes. In fact, the ability to extract $n$-qubits GHZ states and EPR pairs from the original multipartite entangled state constitutes the resource primitives for end-to-end and on-demand quantum communications. Thus, in this paper, we theoretically determine upper and lower bounds for the number of extractable $n$-qubits GHZ states and EPR pairs involving nodes remote in the artificial topology, as well as the achievable size $n$ of remote GHZ states. The theoretical analysis is then complemented by the proposal of a novel algorithm, which provides in polynomial-time a heuristic solution to the above problem. This is remarkable, since the theoretical problem is NP-complete. The performance analysis demonstrates the proposed algorithm is able to effectively manipulate the original and arbitrary graph state for extracting entanglement resources across remote nodes.

On the Efficient Extraction of Entangled Resources

TL;DR

The paper tackles the problem of remotely extracting entangled resources (GHZ states and EPR pairs) from a given two-colorable graph state, where remote nodes are non-adjacent in the artificial topology. It formalizes remote -Gability and remote Pairability, establishes NP-completeness, and provides constructive lower and upper bounds on extractable volumes, including the maximum mass of remote GHZ states. A polynomial-time heuristic, Algorithm Remote Extraction, leverages single-qubit Clifford operations, Pauli measurements, and classical communication to compute volumes, identify node locations, and estimate the maximum GHZ size, with proven polynomial-time complexity. Performance evaluations on bipartite and Internet-inspired graphs demonstrate the algorithm yields nontrivial remote resources (e.g., remote GHZ masses from 3 to 17 and multiple EPR pairs) and scales with graph density. The work lays a foundation for dynamic, end-to-end quantum communications by enabling on-demand remote entanglement extraction, and it discusses future extensions to more general graph classes and noisy environments to enhance practicality in real quantum networks.

Abstract

In the Quantum Internet, multipartite entanglement enables a rich and dynamic overlay topology, referred to as artificial topology, upon the physical one, that can be exploited for communication purposes. In fact, the ability to extract -qubits GHZ states and EPR pairs from the original multipartite entangled state constitutes the resource primitives for end-to-end and on-demand quantum communications. Thus, in this paper, we theoretically determine upper and lower bounds for the number of extractable -qubits GHZ states and EPR pairs involving nodes remote in the artificial topology, as well as the achievable size of remote GHZ states. The theoretical analysis is then complemented by the proposal of a novel algorithm, which provides in polynomial-time a heuristic solution to the above problem. This is remarkable, since the theoretical problem is NP-complete. The performance analysis demonstrates the proposed algorithm is able to effectively manipulate the original and arbitrary graph state for extracting entanglement resources across remote nodes.

Paper Structure

This paper contains 18 sections, 8 theorems, 18 equations, 14 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work And Contribution
  3. Research Problem
  4. Remote Pairability and Remote $n$-Gability
  5. Preliminaries
  6. Remote Extraction Conditions
  7. Algorithm
  8. Algorithm Design
  9. Algorithm Complexity Analysis
  10. Performance Evaluation
  11. General Two-colorable Graph State Performance
  12. Setup
  13. Remote $n$-Gability Performance Analysis
  14. Remote Pairability Performance Analysis
  15. General Graph State Performance
  16. ...and 3 more sections

Key Result

Lemma 1

Let $\ket{G}$ be a two-colorable graph state, with corresponding graph $G=(P_1, P_2, E)$. A sufficient condition for concurrently extracting $\dot{r}_g(n)$ GHZ states, each involving $n$ qubits, is that $\dot{r}_g(n)$ vertices in one partition have pairwise disjoint opposite remote sets of cardinali

Figures (14)

  • Figure 1: Remote vs Vanilla Pairability and Gability for a 5-qubit linear graph state. (a) The initial artificial topology is a 5-qubit linear graph state. (b) Vanilla Pairability allows extraction of up to two EPR pairs from (a). (c) Remote Pairability enables extraction of only one EPR pair between remote nodes from (a). (d) Vanilla Gability permits extraction of a maximal 4-qubit GHZ state from (a). (e) Remote Gability supports extraction of a maximal 3-qubit GHZ state among remote nodes, corresponding to the maximum independent set in (a).
  • Figure 2: Venn diagram for the relationship between GHZ-VM, BELL-VM and REMOTE-VM (our research problem).
  • Figure 3: Pictorial representation of the research problem. (a) The initial 25-qubit bipartite graph state $\ket{G}$. The goal is to constructively address the Remote-VM problem: determining the bound for the volume of extractions can be performed simultaneously from $\ket{G}$, and the bound for the maximum mass of remote GHZ states that can be extracted, and identifying (location) the remote nodes involved in these extractions. (b) A solution to the Remote Pairability problem, identifying pairs of remote nodes that can be entangled. (c–d) Examples for the Remote $n$-Gability problem, with $n=3,4$. (e) Extraction of a 15-qubit remote GHZ state, representing the lower bound of maximum achievable mass of a remote GHZ state from the initial graph. (f) Illustration of diverse extracted remote resources obtained from $\ket{G}$.
  • Figure 4:
  • Figure 5:
  • ...and 9 more figures

Theorems & Definitions (21)

  • Definition 1: Remote Nodes
  • Definition 2: Remote Subnet
  • Definition 3: $\bm{{r_g}(n)}$: remote $\bf{n}$-Gability
  • Remark
  • Definition 4: $\bm{r_e}$: remote Pairability
  • Definition 5: Maximum degree
  • Definition 6: Maximum Independent set
  • Remark
  • Definition 7: Two-colorable Graph or Bipartite Graph
  • Definition 8: Star vertex
  • ...and 11 more