Resource theory of causal connection

Simon Milz1, Jessica Bavaresco1, and Giulio Chiribella2,3,4

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

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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.

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.

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