A de Finetti theorem for quantum causal structures

Fabio Costa; Jonathan Barrett; Sally Shrapnel

doi:10.22331/q-2025-02-11-1628

A de Finetti theorem for quantum causal structures

Fabio Costa^1,2, Jonathan Barrett³, and Sally Shrapnel⁴

¹Nordita, Stockholm University and KTH Royal Institute of Technology, Hannes Alfvéns väg 12 Stockholm, 106 91, Sweden
²School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia
³Quantum Group, Department of Computer Science, University of Oxford
⁴ARC Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia

Published:	2025-02-11, volume 9, page 1628
Editor:	Borivoje Dakic
Eprint:	arXiv:2403.10316v3
Doi:	https://doi.org/10.22331/q-2025-02-11-1628
Citation:	Quantum 9, 1628 (2025).

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Abstract

What does it mean for a causal structure to be `unknown'? Can we even talk about `repetitions' of an experiment without prior knowledge of causal relations? And under what conditions can we say that a set of processes with arbitrary, possibly indefinite, causal structure are independent and identically distributed? Similar questions for classical probabilities, quantum states, and quantum channels are beautifully answered by so-called "de Finetti theorems", which connect a simple and easy-to-justify condition – symmetry under exchange – with a very particular multipartite structure: a mixture of identical states/channels. Here we extend the result to processes with arbitrary causal structure, including indefinite causal order and multi-time, non-Markovian processes applicable to noisy quantum devices. The result also implies a new class of de Finetti theorems for quantum states subject to a large class of linear constraints, which can be of independent interest.

Featured image: Repetitions of an experiment are modelled as single process comprising multiple operations, each performed only once. A de Finetti theorem for processes seeks to group these operations into sets, such that each set represents an independent trial under equivalent conditions, with all trials described by the same, although possibly unknown, process.

Popular summary

In recent years, there has been growing interest in the nature of quantum causal structures—situations where the causal order of events is not fixed but can be uncertain or even fundamentally indefinite. This has led to the development of a general framework to describe such phenomena, which has gained prominence due to its implications for quantum computing, cryptography, and quantum gravity, where spacetime itself may lack a definite causal structure.

Like ordinary quantum mechanics, this framework is fundamentally probabilistic: it predicts the likelihood of different outcomes for sets of measurements, where the causal relations between the measurements themselves can be uncertain or indefinite. To apply the framework, experiments must be repeated multiple times so that predicted probabilities can be compared with observed frequencies. However, this raises a deep question: what does it mean to repeat an experiment when the causal structure is not fixed a priori?

In fact, similar challenges already arise in ordinary probability theory and quantum mechanics: what does it mean to repeat any type of experiment? The key result is de Finetti’s theorem, which states that if a set of observations is exchangeable—meaning their order does not matter and they can be repreated arbitrarily many times—then they can be understood as independent and identically distributed (i.i.d.) trials, up to a general uncertainty about the state associated with each trial. This reduces the question of justifying repetitions to that of ensuring symmetry upon reordering. De Finetti-type theorems also work as technical cornerstones in several applications, ranging from quantum information to many-body physics.

This work extends this idea to quantum processes with arbitrary causal structures. It shows that, regardless of their temporal locations, multiple sets of measurements and operations that are equivalent under reordering can be understood as i.i.d. repetitions of an experiment, in which the order between operations can fluctuate probabilistically or be undefined. The result also extends the applicability of de Finetti-type theorems to scenarios of repeated measurements subject to non-Markovian dynamics, with applications to noisy quantum devices.

► BibTeX data

@article{Costa2025definettitheorem,
  doi = {10.22331/q-2025-02-11-1628},
  url = {https://doi.org/10.22331/q-2025-02-11-1628},
  title = {A de {F}inetti theorem for quantum causal structures},
  author = {Costa, Fabio and Barrett, Jonathan and Shrapnel, Sally},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1628},
  month = feb,
  year = {2025}
}

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[2] Patrick Andriolo, Esteban Vasquez, Elizabeth Agudelo, Max Riegler, Matej Pivoluska, and Gláucia Murta, "Quantum Key Distribution with Imperfections: Recent Advances in Security Proofs", Brazilian Journal of Physics 56 4, 157 (2026).

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