Understanding cause-effect relationships is a crucial part of the scientific process. As Bell's theorem shows, within a given causal structure, classical and quantum physics impose different constraints on the correlations that are realizable, a fundamental feature that has technological applications. However, in general it is difficult to distinguish the set of classical and quantum correlations within a causal structure. Here we investigate a method to do this based on using entropy vectors for Tsallis entropies. We derive constraints on the Tsallis entropies that are implied by (conditional) independence between classical random variables and apply these to causal structures. We find that the number of independent constraints needed to characterize the causal structure is prohibitively high such that the computations required for the standard entropy vector method cannot be employed even for small causal structures. Instead, without solving the whole problem, we find new Tsallis entropic constraints for the triangle causal structure by generalizing known Shannon constraints. Our results reveal mathematical properties of classical and quantum Tsallis entropies and highlight difficulties of using Tsallis entropies for analyzing causal structures.
Understanding cause-effect relationships is a crucial part of the scientific process. As Bell's theorem shows, within a given causal structure, classical and quantum physics impose different constraints on the correlations that are realizable, a fundamental feature that has technological applications. However, in general it is difficult to distinguish the set of classical and quantum correlations within a causal structure. Here we investigate a method to do this based on using entropy vectors for Tsallis entropies. We derive constraints on the Tsallis entropies that are implied by (conditional) independence between classical random variables and apply these to causal structures. We find that the number of independent constraints needed to characterize the causal structure is prohibitively high such that the computations required for the standard entropy vector method cannot be employed even for small causal structures. Instead, without solving the whole problem, we find new Tsallis entropic constraints for the triangle causal structure by generalizing known Shannon constraints. Our results reveal mathematical properties of classical and quantum Tsallis entropies and highlight difficulties of using Tsallis entropies for analyzing causal structures.
A causal structure is a relationship between observed variables that in general restricts the possible correlations between them. This relationship can be mediated by unobserved systems, modelled by random variables in the classical case or joint quantum systems in the quantum case. One way to differentiate between the correlations realisable by two different causal structures is to use entropy vectors, i.e., vectors whose components correspond to the entropies of each subset of the observed variables. To date, the starting point for deriving entropic constraints within causal structures are the so-called Shannon inequalities (positivity of entropy, conditional entropy and conditional mutual information). In the present work we investigate what happens when non-Shannon entropic inequalities are included as well. We show that in general these lead to tighter outer approximations of the set of realisable entropy vectors and hence enable a sharper distinction of different causal structures. Since non-Shannon inequalities can only be applied amongst classical variables, it might be expected that their use enables an entropic distinction between classical and quantum causal structures. However, this remains an open question. We also introduce techniques for deriving inner approximations to the allowed sets of entropy vectors for a given causal structure. These are useful for proving tightness of outer approximations or for finding interesting regions of entropy space. We illustrate these techniques in several scenarios, including the triangle causal structure.
A causal structure is a relationship between observed variables that in general restricts the possible correlations between them. This relationship can be mediated by unobserved systems, modelled by random variables in the classical case or joint quantum systems in the quantum case. One way to differentiate between the correlations realisable by two different causal structures is to use entropy vectors, i.e., vectors whose components correspond to the entropies of each subset of the observed variables. To date, the starting point for deriving entropic constraints within causal structures are the so-called Shannon inequalities (positivity of entropy, conditional entropy and conditional mutual information). In the present work we investigate what happens when non-Shannon entropic inequalities are included as well. We show that in general these lead to tighter outer approximations of the set of realisable entropy vectors and hence enable a sharper distinction of different causal structures. Since non-Shannon inequalities can only be applied amongst classical variables, it might be expected that their use enables an entropic distinction between classical and quantum causal structures. However, this remains an open question. We also introduce techniques for deriving inner approximations to the allowed sets of entropy vectors for a given causal structure. These are useful for proving tightness of outer approximations or for finding interesting regions of entropy space. We illustrate these techniques in several scenarios, including the triangle causal structure.
This thesis reports progress in two domains, causal structures and microscopic thermodynamics, both of which are pertinent in the development of quantum technologies. The first part is dedicated to the analysis of causal structure, which encodes the relationship between observed variables, in general restricting the set of possible correlations between them. Our considerations rely on a recent entropy vector method. Based on this, we develop techniques for deriving entropic constraints to differentiate between causal structures. We provide sufficient conditions for entropy vectors to be realisable within a causal structure and derive new, improved necessary conditions in terms of so-called non-Shannon inequalities. We also report that for a family of causal structures, including the bipartite Bell scenario and the bilocal causal structure, entropy vectors are unable to distinguish between classical and quantum causes, in spite of the existence of quantum correlations that are not classically reproducible. Hence, further development is needed in order to understand cause from a quantum perspective. In the second part we explore an axiomatic framework for modelling error-tolerant processes in microscopic thermodynamics. Our axiomatisation allows for the accommodation of finite precision levels, crucial for describing experiments in the microscopic regime. It leads to the emergence of manageable quantities that give insights into the feasibility and expenditure of processes, which for adiabatic processes are shown to be smooth entropy measures. Our framework also leads to thermodynamic behaviour at the macroscopic scale, meaning that for thermodynamic equilibrium states a unique function provides necessary and sufficient conditions for state transformations, like in the traditional second law.
This thesis reports progress in the analysis of causality and multi-agent logical paradoxes in quantum and post-quantum theories. These research areas are highly relevant for the foundations of physics as well as the development of quantum technologies. In the first part, focussing on causality, we develop techniques for using generalised entropies to analyse distinctions between classical and non-classical causal structures. We derive new properties of Tsallis entropies of systems that follow from the relevant causal structure, and apply these to obtain new necessary constraints for classicality in the Triangle causal structure. Supplementing the method with the post-selection technique, we provide evidence that Shannon and Tsallis entropic constraints are insufficient for detecting non-classicality in Bell scenarios with non-binary outcomes. This points to the need for better methods of characterising correlations in non-classical causal structures. Further, we investigate the relationships between causality and space-time by developing a framework for modelling cyclic and fine-tuned influences in non-classical theories. We derive necessary and sufficient conditions for such causal models to be compatible with a space-time structure and for ruling out operationally detectable causal loops. In particular, this provides an operational framework for analysing post-quantum theories admitting jamming non-local correlations. In the second part, we investigate multi-agent logical paradoxes such as the Frauchiger-Renner paradox and develop a framework for analysing such paradoxes in arbitrary physical theories. Applying this to box world, a post-quantum theory, we derive a stronger paradox that does not rely on post-selection. Our results reveal that reversible evolution of agents' memories is not necessary for deriving multi-agent paradoxes, and that certain forms of contextuality might be.
Causality and logic are both fundamental to our understanding of the universe, but our intuitions about these are challenged by quantum phenomena. This thesis reports progress in the analysis of causality and multi-agent logical paradoxes in quantum and post-quantum theories. Both these research areas are highly relevant for the development of quantum technologies such as quantum cryptography and computing. \n \nPart I of this thesis focuses on causality. Firstly, we develop techniques for using generalised entropies \nto analyse distinctions between classical and non-classical causal structures. We derive new properties of classical and quantum Tsallis entropies of systems that follow from the relevant causal structure, and apply these to obtain new necessary constraints for classicality in the Triangle causal structure. Supplementing the method with the post-selection technique, we provide evidence that Shannon and Tsallis entropic constraints are insufficient for detecting non-classicality in Bell scenarios with non-binary outcomes. This points to the need for better methods of characterising correlations in non-classical causal structures. Secondly, we investigate the relationships between causality and space-time by developing a framework for modelling cyclic and fine-tuned influences in non-classical theories. We derive necessary and sufficient conditions for such causal models to be compatible with a space-time structure and for ruling out operationally detectable causal loops. In particular, this provides an operational framework for analysing post-quantum theories admitting jamming non-local correlations. \n \nIn Part II of this thesis, we investigate multi-agent logical paradoxes, of which the quantum Frauchiger-Renner paradox has been the only example. We develop a framework for analysing such paradoxes in arbitrary physical theories. Applying this to box world, a post-quantum theory, we derive a stronger paradox that does not rely on post-selection. Our results reveal that reversible, unitary evolution of agents' memories is not necessary for deriving multi-agent logical paradoxes, rather that certain forms of contextuality could be.
Causal structures give us a way to understand the origin of observed correlations. These were developed for classical scenarios, but quantum mechanical experiments necessitate their generalisation. Here we study causal structures in a broad range of theories, which include both quantum and classical theory as special cases. We propose a method for analysing differences between such theories based on the so-called measurement entropy. We apply this method to several causal structures, deriving new relations that separate classical, quantum and more general theories within these causal structures. The constraints we derive for the most general theories are in a sense minimal requirements of any causal explanation in these scenarios. In addition, we make several technical contributions that give insight for the entropic analysis of quantum causal structures. In particular, we prove that for any causal structure and for any generalised probabilistic theory, the set of achievable entropy vectors form a convex cone.