Quantum network-entanglement measures

Zhen-Peng Xu; Julio I. de Vicente; Liang-Liang Sun; Sixia Yu

doi:10.22331/q-2025-05-06-1736

Quantum network-entanglement measures

Zhen-Peng Xu¹, Julio I. de Vicente^2,3, Liang-Liang Sun⁴, and Sixia Yu^4,5

¹School of Physics and Optoelectronics Engineering, Anhui University, 230601 Hefei, China
²Departamento de Matemáticas, Universidad Carlos III de Madrid, E-28911, Leganés (Madrid), Spain
³Instituto de Ciencias Matemáticas (ICMAT), E-28049, Madrid, Spain
⁴Department of Modern Physics and National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
⁵Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China

Published:	2025-05-06, volume 9, page 1736
Editor:	Nicolai Friis
Eprint:	arXiv:2311.13945v3
Doi:	https://doi.org/10.22331/q-2025-05-06-1736
Citation:	Quantum 9, 1736 (2025).

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Abstract

Quantum networks are of high interest nowadays and a quantum internet has been long envisioned. Network-entanglement adapts the notion of entanglement to the network scenario and network-entangled states are considered to be a resource to overcome the limitations of a given network structure. In this work, we introduce measures of quantum network-entanglement that are well-defined within the general framework of quantum resource theories, which at the same time have a clear operational interpretation characterizing the extra resources necessary to prepare a targeted quantum state within a given network. In particular, we define the network communication cost and the network round complexity, which turn out to be intimately related to graph-theoretic parameters. We also provide methods to estimate these measures by introducing novel witnesses of network-entanglement.

Featured image: Different rules for classical communication with the network-entanglement measures to be the average time to prepare the target state under those rules, illustrated by a network with 4 vertices two bipartite hyperedges (lines) and one tripartite hyperedge (shadowed rounded triangle), where the arrows indicate the communication direction of classical information after parties implement local operations. In rule (a) only one party implements a local quantum channel and sends classical information to all others. In rule (b) few parties implement local quantum channels and send classical information to their neighbors under the constraint that the senders cannot be receivers. In rule (c) all parties implement local quantum channels and broadcast classical information. The classical information assist the parties to implement local quantum channels.

Popular summary

The notion of quantum networks has become a focus of research, promising superior performance over their classical counterparts in the efficiency of communication, computation, and sensing, with quantum entanglement to be a crucial resource. However, quantum entanglement specified to the network scenario has been considered only very recently, and the resource theory of quantum network-entanglement remains yet unexplored. In this work, we investigate quantum network-entanglement measures to establish the framework of resource theory in the network scenario. We propose the properties the measures should meet and provide explicit constructions of physically motivated and mathematically consistent ones. Since the calculation of these measures is hard (as it is usually the case in quantum resource theories), we provide as well lower and upper bounds using graph-theoretic parameters, which we show to be tight in many relevant cases. Furthermore, we also develop methods to estimate these measures from experimental data.

► BibTeX data

@article{Xu2025quantumnetwork,
  doi = {10.22331/q-2025-05-06-1736},
  url = {https://doi.org/10.22331/q-2025-05-06-1736},
  title = {Quantum network-entanglement measures},
  author = {Xu, Zhen-Peng and Vicente, Julio I. de and Sun, Liang-Liang and Yu, Sixia},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1736},
  month = may,
  year = {2025}
}

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