An introduction to causality in quantum theory

Two projects were completed as part of this Masters thesis: 1. Composable security in relativistic quantum cryptography : The research carried out as part of the thesis on this topic lead to the original research paper [1]. An abstract of this is provided in the preface of this report and will not be discussed further in this report. We refer the reader to the original paper for further information in this regard. 2. Causality in quantum theory and beyond : This will be the topic of this thesis. It consists of a review of quantum causality followed by a detailed analysis of and discussion about two mathematical frameworks for studying quantum causality. Below, we give an abstract of what one will encounter in this report. In the light of Bell’s theorem [2] and its experimental, loophole-free tests [3–5], it has become apparent that the classical understanding of causality [6,7] does not provide satisfactory explanations of quantum experiments involving entangled systems [8]. This implies that the notions of cause in quantum and more general theories are fundamentally different from the classical case. There are several frameworks for studying quantum causality and they can be grouped under two broad categories. One approach is to generalise existing classical causal models to quantum ones [9–13]. Another approach is to start by defining general frameworks for causal structures [14–18], and then narrowing down to identify properties of quantum causal structures. For example, traditionally, one assumes a fixed background space-time structure while describing physical phenomena but recent approaches to modelling causal structures [14–17] within more general frameworks have shown that one can study cause-effect relations even without referring to an underlying space-time structure. This allows one to consider causal structures that are themselves subject to fundamental quantum principles such as superposition and uncertainty and are not compatible with a definite causal order. Some indefinite causal structures of this kind may be seen to arise in situations where quantum mechanics and general relativity are jointly applied [19] and a deeper understanding of these may be crucial for a theory of quantum gravity. In this thesis, we will focus mainly on two frameworks within the second approach: the causal box [18] and the process matrix frameworks [16]. While the process matrix framework models “superpositions of causal orders” in the absence of a global order such as a fixed background spacetime, the causal boxes framework assumes a fixed global order. The quantum switch [20] which has been physically implemented [21,22] is often claimed to be superposition of causal orders but can be represented as a causal box [18] and is hence compatible with a fixed background space-time. This raises the question of whether these are really “superpositions of causal orders” of space-time events or merely “superpositions of the temporal orders” of operations. By analysing the quantum switch in the two frameworks, we argue that the quantum switch as currently implemented does not constitute a superposition of causal orders but rather a (controlled) superposition of temporal orders of fixed operations. To this effect, we provide the mathematical mapping from the causal box representation to the process matrix representation of the quantum switch and discuss the physical differences between them. This also gives interesting insights into the current implementation of the quantum switch and it’s gravitational version [19] which is a proposed thought experiment with no known physical implementations. We then compare the two frameworks more generally and conclude with some interesting open questions in this direction. 1In this version of the thesis, an incorrect conclusion made in one of the discussions of the original version has been corrected and some typos and grammatical errors fixed. The main scientific content and most of the remaining writing is identical to the original version submitted in September 2017. More concretely, the main changes have been made in Section 4.1. 4 where it was earlier mentioned that the causal box and process matrix representations are mathematically equivalent up to vacuum states but this was not the correct conclusion of our mathematical derivation and has now been corrected and this discussion partially rewritten.