Is there causation in fundamental physics? New insights from process matrices and quantum causal modelling

Abstract

In this article we set out to understand the significance of the process matrix formalism and the quantum causal modelling programme for ongoing disputes about the role of causation in fundamental physics. We argue that the process matrix programme has correctly identified a notion of ‘causal order’ which plays an important role in fundamental physics, but this notion is weaker than the common-sense conception of causation because it does not involve asymmetry. We argue that causal order plays an important role in grounding more familiar causal phenomena. Then we apply these conclusions to the causal modelling programme within quantum foundations, arguing that since no-signalling quantum correlations cannot exhibit causal order, they should not be analysed using classical causal models. This resolves an open question about how to interpret fine-tuning in classical causal models of no-signalling correlations. Finally we observe that a quantum generalization of causal modelling can play a similar functional role to standard causal reasoning, but we emphasize that this functional characterisation does not entail that quantum causal models offer novel explanations of quantum processes.

Appendix A: Time reversal

Our description of time reversal has been quite schematic, so let us now be more precise. In fact the technical details of how to implement an appropriate reversal operation are non-trivial, because this requires making a clean separation between the time-symmetric underlying physics and the asymmetries introduced by assumptions about the role of agents, and these features can be difficult to distinguish. Currently, it is known that at least some causal processes which occur in nature are time-reversal invariant in a sense made precise by Pienaar (2019). Given an IS description of a quantum process, Pienaar shows how to write down the corresponding OS process using symmetric informationally complete (SIC) quantum instruments, and then proves that if the directed acyclic graph G associated with the process in the IS description has a certain structure (a ‘layered’ structure) and the process matrix W associated with the IS description is unbiased (inputting the maximally-mixed-state to the process produces the maximally-mixed state as output), then the process is causally reversible, i.e. the OS process is compatible with another IS description whose graph is obtained by reversing the direction of all the arrows in G. That is to say, for at least some causal processes it is indeed the case that we can reverse the IS description to obtain an alternative IS description which corresponds to the same underlying OS process, and moreover this reversed description will still be causal.

Now, it probably is not the case that all causal processes are reversible in this way. For the procedure of simply reversing the direction of the arrows relies on the assumption that, if $O_1$ depends on $I_1$, then after the reversal when $O_1$ becomes $I_2$ and $I_1$ becomes $O_2$, $O_2$ will now depends on $I_2$—and this is not true for all possible dependence relations. For example, suppose we have a map $f(A, B) \rightarrow (C = A, D = A + B)$: clearly here D depends on A. But if we consider the inverse transformation, $f(C, D) \rightarrow (A = C, B = D - C)$ we find that D and A are not connected—so D depends on A but A does not depend on D (thanks to Carlo Rovelli for this example). Thus if we are dealing with processes with dependence relations of this kind, simply reversing all the arrows may not give a correct causal order for the corresponding reversed IS description. However, it is notable that these kinds of dependence relations never occur in classical or quantum physics: within a classical or quantum description, if $O_1$ depends on $I_1$, then after the reversal it will always be the case that $O_2$ depends on $I_2$. So it seems plausible that all processes actually occuring in nature can be reversed in the way that we have described. In particular, we know that all physically possible quantum processes can be obtained from ‘pure’ processes by tracing out an ancilla system (Araújo et al., 2017b), and ‘pure’ processes are those which induce a unitary (and hence reversible) transformation from the past to the future whenever unitary transformations are also applied in the local laboratories; so all quantum processes can indeed be derived from time-reversal invariant underlying physics, and therefore it would seem reasonable to conjecture that all quantum processes should be reversible according to an appropriate formulation of operational time reversal, or at least can be extended using an ancilla to a process which is reversible in this way. This provides further evidence for the idea that fundamental physics does not contain asymmetric causal relations, so causal order is the appropriate way of describing its structure.

That said, we emphasise that our argument does not depend on the conjecture that all physically possible processes are reversible in this way. Our aim here is to address the question of whether there can be causation in fundamental physics even if it is the case that, as many physicists currently believe, fundamental physics contains no intrinsic temporal asymmetries. So the point we want to make is simply that the notion of ‘causality’ employed in the definition of ‘causal order’ and ‘causal process’ does not need an underlying asymmetry: processes may instantiate a well-defined causal order regardless of whether or not they can be reversed to yield another causally ordered process. Pienaar’s result demonstrates that at least some causal processes can indeed be reversed in this way, thus verifying that the definition of ‘causal’ does not depend in any crucial way on asymmetry.