Quantum networks theory

Pablo Arrighi; Amélia Durbec; Matt Wilson

doi:10.22331/q-2024-10-23-1508

Quantum networks theory

Pablo Arrighi¹, Amélia Durbec², and Matt Wilson^3,4,5

¹Université Paris-Saclay, Inria, CNRS, LMF, 91190 Gif-sur-Yvette, France
²CNRS, Centrale Lille, JUNIA, Univ. Lille, Univ. Valenciennes, IEMN, 59046 Lille Cedex, France
³PPLV Group, Department of Computer Science, University College London
⁴Quantum Group, Department of Computer Science, University of Oxford, UK
⁵HKU-Oxford Joint Laboratory for Quantum Information and Computation

Published:	2024-10-23, volume 8, page 1508
Eprint:	arXiv:2110.10587v3
Doi:	https://doi.org/10.22331/q-2024-10-23-1508
Citation:	Quantum 8, 1508 (2024).

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Abstract

The formalism of quantum theory over discrete systems is extended in two significant ways. First, quantum evolutions are generalized to act over entire network configurations, so that nodes may find themselves in a quantum superposition of being connected or not, and be allowed to merge, split and reconnect coherently in a superposition. Second, tensors and traceouts are generalized, so that systems can be partitioned according to almost arbitrary logical predicates in a robust manner. The hereby presented mathematical framework is anchored on solid grounds through numerous lemmas. Indeed, one might have feared that the familiar interrelations between the notions of unitarity, complete positivity, trace-preservation, non-signalling causality, locality and localizability that are standard in quantum theory be jeopardized as the neighbourhood and partitioning between systems become both quantum, dynamical, and logical. Such interrelations in fact carry through, albeit two new notions become instrumental: consistency and comprehension.

Featured image: A fully quantum superposition of networks

Popular summary

Quantum evolutions can place objects in strange situations of being, say, both left and right with some amplitude. They could, in principle, also act on the geometry of things, e.g. placing Alice and Bob in a quantum superposition of being neighbours, or not. In which case they may or may not be able to communicate, depending in the branch of the quantum superposition.

This paper provides the adequate mathematical formalism to reason about those situations.

► BibTeX data

@article{Arrighi2024quantumnetworks,
  doi = {10.22331/q-2024-10-23-1508},
  url = {https://doi.org/10.22331/q-2024-10-23-1508},
  title = {Quantum networks theory},
  author = {Arrighi, Pablo and Durbec, Am{\'{e}}lia and Wilson, Matt},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1508},
  month = oct,
  year = {2024}
}

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Cited by

[1] Phillip S. Isaac, Jon Links, Inna Lukyanenko, and Jason L. Werry, "Explicit conserved operators for a class of integrable bosonic networks from the classical Yang-Baxter equation", Journal of Geometry and Physics 225, 105837 (2026).

[2] Matt Wilson, Giulio Chiribella, and Aleks Kissinger, "Quantum Supermaps are Characterized by Locality", Quantum 10, 2013 (2026).

[3] Theophanes Raptis and Vasilios Raptis, The 5th International Conference on Symmetry (Symmetry 2025) 5 (2025).

[4] Matt Wilson and Augustin Vanrietvelde, "Composable constraints", arXiv:2112.06818, (2021).

[5] Kathleen Barsse, Paolo Perinotti, Alessandro Tosini, and Leonardo Vaglini, "Causal influence versus signaling for interacting quantum channels", Physical Review Research 6 4, 043305 (2024).

[6] Pablo Arrighi, Christopher Cedzich, Marin Costes, Ulysse Rémond, and Benoît Valiron, "Addressable quantum gates", arXiv:2109.08050, (2021).

[7] Pablo Arrighi, Amélia Durbec, and Aurélien Emmanuel, "Size-varying reversible causal graph dynamics", arXiv:1805.10330, (2018).

[8] Augustin Vanrietvelde, Octave Mestoudjian, and Pablo Arrighi, "Partitions in quantum theory", arXiv:2506.22218, (2025).

[9] Phillip S. Isaac, Jon Links, Inna Lukyanenko, and Jason L. Werry, "Explicit conserved operators for a class of integrable bosonic networks from the classical Yang-Baxter equation", arXiv:2507.00483, (2025).

[10] Pablo Arrighi, Amélia Durbec, and Matt Wilson, "Generalised tensors and traces", arXiv:2202.11340, (2022).

[11] Pablo Arrighi, Marios Christodoulou, and Amélia Durbec, "On quantum superpositions of graphs, no-signalling and covariance", Journal of Physics A Mathematical General 58 15, 155303 (2025).

[12] Pablo Arrighi, Marin Costes, and Luidnel Maignan, "Space-time reversible graph rewriting", arXiv:2510.03296, (2025).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-17 21:45:47) and SAO/NASA ADS (last updated successfully 2026-08-17 21:45:47). The list may be incomplete as not all publishers provide suitable and complete citation data.

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