Quantum causal structure

Thesis

Quantum theory challenges our intuitions of causality, as manifest in the existence of Bell non-local correlations. The latter seem to falsify Reichenbach’s common cause principle and evade causal explanations within the classical causal model framework developed in recent decades. But could quantum correlations be explained in causal, albeit quantum causal terms? Even if one embraces a revised causal relation, challenges of our intuitions seem to persist — for instance, ‘quantum indefinite causal order’ of events being conceivable and intensely studied.

This thesis hopes to contribute to our understanding of how causal reasoning can be maintained, in a rigorous manner, in light of quantum theory. Inspired by the work of Allen et al. [Phys. Rev. X 7, 031021 (2017)], a definition of quantum causal relations is given in terms of influence in underlying unitary transformations. The notion of quantum causal structure that ensues is then explored in three, closely related directions.

First, a quantum causal model framework is presented that generalises classical causal models and allows causal explanations of quantum processes, assuming that a definite causal order does exist. Amongst other things, notions of ‘quantum conditional independence’ are presented and generalisations of core classical theorems, such as the d-separation theorem, are derived.

Second, the thesis studies how causal structure of unitary transformations can be understood in terms of their compositional structure. The results here reveal how causal structure is closely associated with the interplay between direct sums and direct products of Hilbert spaces. An extension of quantum circuit diagrams is introduced to visualise the found ‘causal decompositions’ of unitaries.

Third, a generalisation of quantum causal models to cyclic causal structure is proposed, to analyse processes that feature indefinite causal order. The idea is illustrated through analysing well-known examples of such processes and some first results are presented.

Authors: Lorenz, R

• DOI: 10.5287/ora-2nedjjdex
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: causality; quantum causal relations; causal model frameworks
• Subjects: Quantum theory