Causal structure in the presence of sectorial constraints, with application to the quantum switch

Nick Ormrod1, Augustin Vanrietvelde1,2,3, and Jonathan Barrett1

1Quantum Group, Department of Computer Science, University of Oxford
2Department of Physics, Imperial College London
3HKU-Oxford Joint Laboratory for Quantum Information and Computation

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Abstract

Existing work on quantum causal structure assumes that one can perform arbitrary operations on the systems of interest. But this condition is often not met. Here, we extend the framework for quantum causal modelling to situations where a system can suffer $\textit{sectorial constraints}$, that is, restrictions on the orthogonal subspaces of its Hilbert space that may be mapped to one another. Our framework (a) proves that a number of different intuitions about causal relations turn out to be equivalent; (b) shows that quantum causal structures in the presence of sectorial constraints can be represented with a directed graph; and (c) defines a fine-graining of the causal structure in which the individual sectors of a system bear causal relations. As an example, we apply our framework to purported photonic implementations of the quantum switch to show that while their coarse-grained causal structure is cyclic, their fine-grained causal structure is acyclic. We therefore conclude that these experiments realize indefinite causal order only in a weak sense. Notably, this is the first argument to this effect that is not rooted in the assumption that the causal relata must be localized in spacetime.

In science and in everyday life, we very commonly explain things using the concepts of cause and effect. When we see many puddles in the street, we assume they are all effects of the same cause — the rain. When we encourage people to quit smoking, it is because we believe it causes cancer.

And yet our most successful scientific theory — quantum theory — suggests our most basic ideas about causation and causal reasoning are somehow mistaken. The famous nonlocal correlations that violate Bell's inequalities resist causal explanation as traditionally understood, and the possibility of putting objects into superpositions seems to allow for situations in which there is no definite fact about the direction of causal influence.

As a result, there has been much effort in recent years to modify our causal notions for a quantum setting. Our paper extends the study of intrinsically quantum causal structures to a new range of scenarios. One of the consequences is that recent experiments that aim to create an indefinite direction of causal influence can be understood as "weakly" indefinite — even more strongly indefinite directions of influence are conceivable.

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