An entropic approach to quantum causal structures

The notions of cause and effect are fundamental in physics, as well as in all natural sciences. In recent years, methods of inference of causal relations directly from observed data have been developed by researcher in statistics and artificial intelligence. Quantum probabilities and statistical data collected in quantum experiments, however, seem to radically differ from their classical analogues, thus challenging our intuition on cause-and-effect relations. This project aims to develop a consistent theory of quantum causal structure that captures such nonclassical aspects of quantum theory. In particular, we aim to extend the information theoretic approach, based on the notion of entropy, from the classical to the quantum domain, and possibly beyond it, to general probabilistic theories. The project will provide analytical and computational tools for the investigation of causal structures able to characterize the entropy regions associated with a given causal structure, hence, able to discriminate between data consistent or inconsistent with the assumed causal relations.

The notions of cause and effect are fundamental in physics, as well as in all natural sciences. In recent years, methods of inference of causal relations directly from observed data have been developed by researcher in statistics and artificial intelligence. Quantum probabilities and statistical data collected in quantum experiments, however, seem to radically differ from their classical analogues, thus challenging our intuition on cause-and-effect relations. This project aimed to develop a consistent theory of quantum causal structure that captures such nonclassical aspects of quantum theory, in particular, to extend the information theoretic approach, based on the notion of entropy, from the classical to the quantum domain, and possibly beyond it, to general probabilistic theories. The project provided analytical and computational tools for the investigation of causal structures able to characterize the entropy regions associated with a given causal structure, hence, able to discriminate between data consistent or inconsistent with the assumed causal relations. The project explored various aspects of causality from the perspective of quantum information theory. First, we developed of a consistent language to describe causality and causal correlations, based on the notion of entropy, and applicable both in the context of classical and quantum theory. We developed algorithms and software for characterising entropy regions associated with different causal scenarios. We showed that our formalism is able to detect, via causal entropic inequalities, correlation outside the set of causal ones, possibly generated by quantum superposition of causal orders. Motivated by the interpretation of such correlations, or processes, as a resource for quantum information processing shared by two (or more) parties, we investigate the notion of composition of processes. We show that under very basic assumptions such a composition rule does not exist. While the availability of multiple independent copies of a resource, e.g. quantum states or channels, is the starting point for defining information-theoretic notions such as entropy (both in classical and quantum information theory), our no-go result means that an information theory of general quantum processes will not possess a natural rule for the composition of resources. Another aspect of our project was the investigation of definite causal order: we explored quantum correlations both in the temporal scenario, i.e., events able to directly influence one another, and in the spatial one, i.e., events unable to influence each other, but with a common past. In addition, we explored their connection to the problem of of measurement incompatibility, i.e., the impossibility of jointly measure some properties of a quantum system. Several results have been obtained, in particular, on the characterisation of such correlations, the quantification of incompatibility and detection of incompatibility structures in a device-independent framework, i.e., without assumptions on the functioning of physical devices used in experimental tests, and the role of measurement incompatibility in the generation of nonclassical temporal correlations.