Causal influence in operational probabilistic theories

TL;DR

This paper explores the concept of causal influence in operational probabilistic theories, revealing that influence can exist without signalling, and introduces conditions where they coincide, impacting the understanding of causal networks and cellular automata.

Contribution

It introduces a new, weaker definition of causal influence applicable beyond quantum theory, and clarifies the relationship between influence and signalling in classical and quantum contexts.

Findings

Causal influence can occur without signalling, even in classical theories.

The new definition redefines neighbourhoods in classical cellular automata.

Under no interaction without disturbance, signalling and causal influence are equivalent.

Abstract

We study the relation of causal influence between input systems of a reversible evolution and its output systems, in the context of operational probabilistic theories. We analyse two different definitions that are borrowed from the literature on quantum theory -- where they are equivalent. One is the notion based on signalling, and the other one is the notion used to define the neighbourhood of a cell in a quantum cellular automaton. The latter definition, that we adopt in the general scenario, turns out to be strictly weaker than the former: it is possible for a system to have causal influence on another one without signalling to it. Remarkably, the counterexample comes from classical theory, where the proposed notion of causal influence determines a redefinition of the neighbourhood of a cell in cellular automata. We stress that, according to our definition, it is impossible anyway to have causal influence in the absence of an interaction, e.g.~in a Bell-like scenario. We study various conditions for causal influence, and introduce the feature that we call no interaction without disturbance, under which we prove that signalling and causal influence coincide. The proposed definition has interesting consequences on the analysis of causal networks, and leads to a revision of the notion of neighbourhood for classical cellular automata, clarifying a puzzle regarding their quantisation that apparently makes the neighbourhood larger than the original one.