# Causal influence in operational probabilistic theories

Paolo Perinotti

QUIT Group, Dipartimento di Fisica, Università degli studi di Pavia, and INFN sezione di Pavia, via Bassi 6, 27100 Pavia, Italy

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### 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 $\textit{no interaction without disturbance}$, 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.

The paper addresses the question as to whether a given reversible evolution of the state of a network of systems produces an influence from a node to another one. The starting tool to face this problem is to define in precise terms what we mean by causal influence. Usually, the literature on the subject identifies such an influence with the ability to use the evolution to send information from the first node to the second one. There is however a second notion, developed within the theory of quantum cellular automata, that identifies such an influence with the propagation of the effects of a local intervention at the first node to the second one. The paper adopts the second point of view, and shows that this definition is actually wider: one can have this second kind of causal influence from the first node without the possibility of sending information to the second node. What sort of influence is there when we are not allowed to exploit it for communication purposes? The answer is remarkably simple:
if we can create correlations between the second node and the first one, there is a causal influence on the second node, even without affecting its local state.
This is the case in classical information theory, where we can copy a bit without the need of affecting it.

This result has an impact on the analysis of causal relations within causal networks, like, for example, in the definition of the neighbourhood of a cell in a cellular automaton, and its cone of causal influence—generalising the notion of a light cone in the spacetime of relativity.

The analysis is carried out in a wide scenario of information theories, that comprises those of classical and quantum systems as a special case, but also embraces more exotic post-quantum theories.

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