Analysing causal structures in generalised probabilistic theories

Mirjam Weilenmann1,2 and Roger Colbeck2

1Institute for Quantum Optics and Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, AT
2Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK

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Abstract

Causal structures give us a way to understand the origin of observed correlations. These were developed for classical scenarios, but quantum mechanical experiments necessitate their generalisation. Here we study causal structures in a broad range of theories, which include both quantum and classical theory as special cases. We propose a method for analysing differences between such theories based on the so-called measurement entropy. We apply this method to several causal structures, deriving new relations that separate classical, quantum and more general theories within these causal structures. The constraints we derive for the most general theories are in a sense minimal requirements of any causal explanation in these scenarios. In addition, we make several technical contributions that give insight for the entropic analysis of quantum causal structures. In particular, we prove that for any causal structure and for any generalised probabilistic theory, the set of achievable entropy vectors form a convex cone.

Quantum theory has many strange features that set it apart from the classical theory before it. Perhaps the most striking is the 'spooky action at a distance', where a quantum measurement on one particle leads to a state update on another that appears to go against the spirit of relativity theory, although without allowing superluminal signalling. In the 1960s Bell pinned this down more precisely by showing that describing quantum correlations in a causal model requires either dropping the notion of locality, or modifying the usual definition of causation.

In this work we consider modified notions of causation within a broad class of theories (generalised probabilistic theories) that include classical and quantum theory as special cases but also theories whose correlations are more 'spooky' than those in quantum mechanics. We show how to derive causal constraints within such theories, and apply our technique to a variety of causal structures. By studying causation in the most general post-quantum theory, we derive constraints that must hold for any causal explanations of correlations, and so can be understood as minimal requirements of causation itself. Furthermore, by comparing general constraints to those in quantum theory provides a way to investigate what is special about quantum mechanics.

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[2] V. Vilasini and Roger Colbeck, "Analyzing causal structures using Tsallis entropies", Physical Review A 100 6, 062108 (2019).

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