Knot invariants and indefinite causal order

Samuel Fedida; Anne-Catherine de la Hamette; Viktoria Kabel; Časlav Brukner

doi:10.22331/q-2025-10-06-1875

Knot invariants and indefinite causal order

Samuel Fedida¹, Anne-Catherine de la Hamette^2,3, Viktoria Kabel^3,2, and Časlav Brukner^2,3

¹Centre for Quantum Information and Foundations, DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
²University of Vienna, Faculty of Physics, Vienna Doctoral School in Physics, and Vienna Center for Quantum Science and Technology (VCQ), Boltzmanngasse 5, A-1090 Vienna, Austria
³Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria

Published:	2025-10-06, volume 9, page 1875
Editor:	John van de Wetering
Eprint:	arXiv:2409.11448v2
Doi:	https://doi.org/10.22331/q-2025-10-06-1875
Citation:	Quantum 9, 1875 (2025).

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Abstract

We explore indefinite causal order between events in the context of quasiclassical spacetimes in superposition. We introduce several new quantifiers to measure the degree of indefiniteness of the causal order for an arbitrary finite number of events and spacetime configurations in superposition. By constructing diagrammatic and knot-theoretic representations of the causal order between events, we find that the definiteness or maximal indefiniteness of the causal order is topologically invariant. This reveals an intriguing connection between the field of quantum causality and knot theory. Furthermore, we provide an operational encoding of indefinite causal order and discuss how to incorporate a measure of quantum coherence into our classification.

Featured image: Created by Kaća Bradonjić.

Knot invariants and indefinite causal order – presentation at Perimeter Institute

Popular summary

In classical general relativity, spacetime structure determines the causal order between events. When we allow for the possibility to place spacetimes in quantum superposition, fascinating phenomena emerge. These include processes with "indefinite causal order", where two events happen in a superposition of different orders – A before B and B before A.

In this work, we connect this research area with a seemingly unrelated field: the mathematical theory of knots. We show how to map diagrams representing indefinite causal order to knots, with the knot structure reflecting the definiteness or indefiniteness of the causal order. This connection not only provides an elegant representation of superpositions of causal order but also allows to import established results from knot theory to better understand the structure of processes with indefinite causal order.

► BibTeX data

@article{Fedida2025knotinvariants,
  doi = {10.22331/q-2025-10-06-1875},
  url = {https://doi.org/10.22331/q-2025-10-06-1875},
  title = {Knot invariants and indefinite causal order},
  author = {Fedida, Samuel and Hamette, Anne-Catherine de la and Kabel, Viktoria and Brukner, {\v{C}}aslav},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1875},
  month = oct,
  year = {2025}
}

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Cited by

[1] Jorge Escandón-Monardes, "A map of indefinite causal order", arXiv:2506.04607, (2025).

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