Emergence of causal order in quantum theory and beyond

The notions of cause and effect are fundamental in physics, as well as in all natural sciences, but are strongly challenged by the most recent physical theories, which are based on quantum mechanics. Recent experimental tests with elementary quantum systems, like most notably Bell tests, radically challenge the very notions of physical reality and cause-effect relations. In parallel, borrowing and extending concepts of information theory, physical quantities such as energy, temperature and correlations have recently been connected with notions of complexity of communications and computations, with measures like entropies playing a central role, also in a quantitative investigation of causal influences between parts. In its extreme limits, one can extend quantum mechanics and imagine physical theories where a global causal order among space-time regions is not fixed, while maintaining logical consistency and agreement with observed data. Still, such theories would predict correlations between subsystems which can be tested experimentally, similarly as Bell tests, thereby providing more stringent tests of fundamental causal relations. In this project, we aim to investigate the notion of causality both in quantum and in possible indefinite-order theories, and find out how a definite time-order, or arrow of time, may arise in the framework of indefinite- order theories. In particular, we aim to answer the following questions. How can we quantify the strength of causal influences between time-ordered events? Is it related to information-theoretic notions of complexity of sequences of measurement outcomes? How does this extend to indefinite-order theories? Under what physical principles does the notion of causal order emerge? In addition to the above fundamental questions, we aim to explore the practical implications of our findings, in particular, for information processing and transmission (e.g., computation, cryptography). What are the advantages of indefinite-order theories? What are the resources and the costs of simulating them with definite-order or classical resources? What type of experiments can be designed to test such theories? The originality and innovative aspect of our project resides in its combination of fundamental ideas in physics (causality, time-order) with theoretical computer science (information measures, complexity).

The notions of cause and effect are fundamental in all natural sciences, but are strongly challenged by recent theories in physics. Experimental tests with elementary quantum systems, most notably Bell tests, have radically challenged the very notions of physical reality and cause-effect relations. In this project, we extended concepts of information theory to go beyond quantum mechanics and imagined physical theories where a global causal is not necessarily fixed, while insisting that whatever new theories we create would be logically consistent. Bell non-locality - the study of no-signalling correlations in quantum theory - is, by now, widely reported on. On the other hand, signalling correlations (the emergent patterns in the results arising from communication scenarios) have garnered far less attention. These correlations are also referred to as 'non-causal' because they can demonstrate a connection between two parties that cannot be explained by any combination of totally ordered events. In our research we are the first to provide a theory with a closed form characterisation of a non-trivial set of signalling correlations by employing an information theoretic principle: no signalling without system exchange; a natural requirement that says: you cannot acquire information unless someone physically sends it to you. Our explicit characterisation provides the platform on which other researchers can put their results 'on the map'. We also found that the causal relations among any set of space-time regions can be expressed in terms of graph theory - a mathematical formalism for expressing relations between nodes and their connecting edges. On these graphs, we found a way to track the causal consistencies and inconsistencies of different scenarios and showed that what was believed to be two types of logically incompatible setups (the grandparent paradox and the information antinomy), are actually one in the same thing. We also made progress in understanding information in time: any concrete information processing task can be viewed as the sequential manipulation of information over time, receiving inputs and producing outputs. Different physical theories, classical, quantum,and beyond, provide different constraints on how the information can be manipulated. We explored these limitations from the abstract viewpoint of the generation of correlations (between inputs and outputs) over time. We develop optimization methods for temporal correlations in classical and quantum theories, as well as a notion of complexity for output sequences, which allows us to identify quantum advantages. Beyond these foundational questions, our results found applications in the problem of designing optimal quantum clocks, characterizing the dynamics of open quantum systems, and clarifying the properties of quantum memories. Finally, ideas developed for the solution of these specific problems found a broader application as general optimization techniques for convex problems.