Resource theory of causal connection

TL;DR

This paper develops a comprehensive resource theory framework for quantifying and transforming causal connections in multi-party quantum processes, including both definite and indefinite causal order scenarios.

Contribution

It introduces a unified resource theory of causal connection, characterizes free processes, and establishes measures like signalling robustness, advancing understanding of quantum causal structures.

Findings

Identifies most resourceful processes in bipartite and tripartite cases.

Establishes signalling robustness as a key resource monotone.

Shows the uniqueness of the resource theory of causal non-separability.

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.