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.
Highlights
In the last two decades, studies on the foundations of Quantum Theory flourished, nurtured by the wealth of results in quantum information theory and their impact on the understanding of the quantum realm
Operational Probabilistic Theories (OPTs) are defined as all the possible theories that share with classical or quantum theory some basic structure: in particular, the properties of rules by which one can form composite processes as sequences of other processes, form composite systems, or apply processes independently on subsystems of a composite system, as well as the properties of rules to calculate probabilities of events that can occur as alternative outcomes within a process
One can prove further results about uniqueness of reversible dilations modulo reversible transformations on D, we will consider here only the result discussed in the following subsection, that will turn useful in the remainder, and that highlights the main difference between CT discussed above on one hand, and Quantum Theory (QT) and Fermionic Theory (FT) on the other hand
Summary
In the last two decades, studies on the foundations of Quantum Theory flourished, nurtured by the wealth of results in quantum information theory and their impact on the understanding of the quantum realm. In this case we claim that the perturbation of the electromagnetic field detected at x1 is a consequence of the initial intervention at x0 This way of revealing a causal influence relation between two systems mixes two distinct notions: the first one is based on testing the consequences of a controlled intervention occurred on the first system as they propagate to the second system, and the second one is based on identifying those consequences as the possibility of transmitting information. We decouple these two aspects, and focus on the first one. The question underpinning our definition is the following: if we had to simulate the consequences of a hypothetical intervention on system A occurred before the evolution U , by intervening after U instead, would a non-trivial action on system B be required? If the answer is positive, we will say that the evolution U mediates a causal influence from A to B
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