Resource theory of causal connection

Simon Milz; Jessica Bavaresco; Giulio Chiribella

doi:10.22331/q-2022-08-25-788

Resource theory of causal connection

Simon Milz¹, Jessica Bavaresco¹, and Giulio Chiribella^2,3,4

¹Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
²QICI Quantum Information and Computation Initiative, Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong
³Department of Computer Science, University of Oxford, Wolfson Building, 15 Parks Road, Oxford OX1 3QD, United Kingdom
⁴Perimeter Institute for Theoretical Physics, 31 Caroline St North, Waterloo, ON N2L 2Y5, Canada

Published:	2022-08-25, volume 6, page 788
Eprint:	arXiv:2110.03233v3
Doi:	https://doi.org/10.22331/q-2022-08-25-788
Citation:	Quantum 6, 788 (2022).

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Abstract

The capacity of distant parties to send signals to one another is a fundamental requirement in many information-processing tasks. Such ability is determined by the causal structure connecting the parties, and more generally, by the intermediate processes carrying signals from one laboratory to another. Here we build a fully fledged resource theory of causal connection for all multi-party communication scenarios, encompassing those where the parties operate in a definite causal order and also where the order is indefinite. We define and characterize the set of free processes and three different sets of free transformations thereof, resulting in three distinct resource theories of causal connection. In the causally ordered setting, we identify the most resourceful processes in the bipartite and tripartite scenarios. In the general setting, instead, our results suggest that there is no global most valuable resource. We establish the signalling robustness as a resource monotone of causal connection and provide tight bounds on it for many pertinent sets of processes. Finally, we introduce a resource theory of causal non-separability, and show that it is – in contrast to the case of causal connection – unique. Together our results offer a flexible and comprehensive framework to quantify and transform general quantum processes, as well as insights into their multi-layered causal connection structures.

Popular summary

Communication between distant parties is a cornerstone of many information technologies. Its success depends on the quality of the available communication channels and, more broadly, on the spacetime in which the communicating parties are embedded, which determines the possibilities of signaling among them. In recent years, it has been observed that the combination of quantum mechanics and information theory gives rise to a wider class of communication scenarios than classically conceivable, possibly even allowing for scenarios where the causal structure is indefinite. A fundamental question is then how to quantify the communication resources of the causal structure connecting a set of parties. Here we provide a means of quantification in terms of a resource theory of causal connection, which encompasses all quantum communication networks, both with definite and with indefinite causal order. In addition, we show that the same tools allow one to develop a resource theory of indefinite causality. Together, our results offer versatile tools to analyze the quality of general quantum networks and characterize them in terms of their signalling structures.

► BibTeX data

@article{Milz2022resourcetheoryof,
  doi = {10.22331/q-2022-08-25-788},
  url = {https://doi.org/10.22331/q-2022-08-25-788},
  title = {Resource theory of causal connection},
  author = {Milz, Simon and Bavaresco, Jessica and Chiribella, Giulio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {788},
  month = aug,
  year = {2022}
}

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[5] Paulina Lewandowska, Łukasz Pawela, and Zbigniew Puchała, "Strategies for single-shot discrimination of process matrices", Scientific Reports 13 1, 3046 (2023).

[6] Simon Milz and Marco Túlio Quintino, "Transformations between arbitrary (quantum) objects and the emergence of indefinite causality", arXiv:2305.01247, (2023).

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