In this chapter we introduce the postulate of causality, and derive the mathematical structure of a causal OPT. We show how the present definition of causality is equivalent to the Einsteinian one, and argue that it is the only possible notion once it is stripped from its spurious deterministic connotation. In addition, the notion of causality used here is compliant with the historical philosophical concept since David Hume, and with the idea used in common reasoning, inference, and modeling in human sciences. In brief, the present notion of causality can be stated as the requirement of no signaling from the future . This is also the only requisite for an OPT in order to satisfy the principle of no signaling without interaction . In mathematical terms in an OPT the causality postulate is equivalent to the uniqueness of the deterministic effect, which in turn is equivalent to the ability to normalize states. The notion of causality is so naturally embedded in our understanding of reality that it often remains unconsciously implicit in foundational research, even in recent axiomatizations. Causality: From Cinderella to Principle Causality has been the object of a very extensive literature encompassing hundreds of books and technical articles in a wide spectrum of disciplines, ranging from pure philosophy to economics, law, natural sciences, and obviously, physics. Also due to its involvement in such heterogeneous blend of disciplines, causality has long been a vexed notion. Perhaps the most natural connection between concepts of causality in different branches of knowledge is the one at the borderline between physics and philosophy, since the early work of Aristotle, up to the cornerstone of Renée Descartes, who broke the ground for the modern view of David Hume and Immanuel Kant, through to contemporary works on physical causation. Causality has always remained in the realm of philosophy, staying only in the background of physics, without the status of a “law” or the rank of a “principle.” Most of the time causality creeps into physical theories in the form of ad hoc assumptions based on empirical evidences, as when we discard advanced potentials in electrodynamics or when we motivate the Kramers–Kronig relations. In other cases, causality is embodied in the “interpretation” of the theory, as for the Special Relativity of Einstein.
In this chapter we introduce the postulate of causality, and derive the mathematical structure of a causal OPT. We show how the present definition of causality is equivalent to the Einsteinian one, and argue that it is the only possible notion once it is stripped from its spurious deterministic connotation. In addition, the notion of causality used here is compliant with the historical philosophical concept since David Hume, and with the idea used in common reasoning, inference, and modeling in human sciences. In brief, the present notion of causality can be stated as the requirement of no signaling from the future . This is also the only requisite for an OPT in order to satisfy the principle of no signaling without interaction . In mathematical terms in an OPT the causality postulate is equivalent to the uniqueness of the deterministic effect, which in turn is equivalent to the ability to normalize states. The notion of causality is so naturally embedded in our understanding of reality that it often remains unconsciously implicit in foundational research, even in recent axiomatizations. Causality: From Cinderella to Principle Causality has been the object of a very extensive literature encompassing hundreds of books and technical articles in a wide spectrum of disciplines, ranging from pure philosophy to economics, law, natural sciences, and obviously, physics. Also due to its involvement in such heterogeneous blend of disciplines, causality has long been a vexed notion. Perhaps the most natural connection between concepts of causality in different branches of knowledge is the one at the borderline between physics and philosophy, since the early work of Aristotle, up to the cornerstone of Renée Descartes, who broke the ground for the modern view of David Hume and Immanuel Kant, through to contemporary works on physical causation. Causality has always remained in the realm of philosophy, staying only in the background of physics, without the status of a “law” or the rank of a “principle.” Most of the time causality creeps into physical theories in the form of ad hoc assumptions based on empirical evidences, as when we discard advanced potentials in electrodynamics or when we motivate the Kramers–Kronig relations. In other cases, causality is embodied in the “interpretation” of the theory, as for the Special Relativity of Einstein.
Causality has never gained the status of a 'law' or 'principle' in physics. Some recent literature has even popularized the false idea that causality is a notion that should be banned from theory. Such misconception relies on an alleged universality of the reversibility of the laws of physics, based either on the determinism of classical theory, or on the multiverse interpretation of quantum theory, in both cases motivated by mere interpretational requirements for realism of the theory. Here, I will show that a properly defined unambiguous notion of causality is a theorem of quantum theory, which is also a falsifiable proposition of the theory. Such a notion of causality appeared in the literature within the framework of operational probabilistic theories. It is a genuinely theoretical notion, corresponding to establishing a definite partial order among events, in the same way as we do by using the future causal cone on Minkowski space. The notion of causality is logically completely independent of the misidentified concept of 'determinism', and, being a consequence of quantum theory, is ubiquitous in physics. In addition, as classical theory can be regarded as a restriction of quantum theory, causality holds also in the classical case, although the determinism of the theory trivializes it. I then conclude by arguing that causality naturally establishes an arrow of time. This implies that the scenario of the 'block Universe' and the connected 'past hypothesis' are incompatible with causality, and thus with quantum theory: they are both doomed to remain mere interpretations and, as such, are not falsifiable, similar to the hypothesis of 'super-determinism'.This article is part of a discussion meeting issue 'Foundations of quantum mechanics and their impact on contemporary society'.
[I] think I can safely say that nobody understands quantum mechanics. Richard Feynman, The Character of Physical Law Quantum mechanics [is] that mysterious, confusing discipline, which none of us really understands but which we know how to use. It works perfectly, as far as we can tell, in describing physical reality, but it is a ‘counter-intuitive discipline’… Murray Gell-Mann, Questions for the Future [The quantum] postulate implies a renunciation as regards the causal space-time co-ordination of atomic processes. Niels Bohr, Atomic Theory and the Description of Nature It should be emphasized, however, that the probability function does not in itself represent a course of events in the course of time. It represents a tendency for events and our knowledge of events. The probability function can be connected with reality only if one essential condition is fulfilled: if a new measurement is made to determine a certain property of the system. [T]he idea of an objective real world whose smallest parts exist objectively in the same sense as stones or trees exist, independently of whether or not we observe them…is impossible … Werner Heisenberg, Physics and Philosophy [In] quantum theory it is the principle of causality , or more accurately that of determinism , which must be dropped and replaced by something else.… We now have a new form of the law of causality.… It is as follows: if in a certain process the initial conditions are determined as accurately as the uncertainty relations permit, then the probabilities of all possible subsequent states are governed by exact laws. Max Born, The Restless Universe
Quantum mechanics has achieved unparalleled success as an operational theory, describing a wide range of experiments to remarkable accuracy. However, the physical foundations on which it rests remain as puzzling as they were a century ago, and a concise statement of the physical principles that underlie quantum mechanics is still outstanding. One promising approach holds that these principles should be formulated in terms of information, since many of the counter-intuitive effects in quantum theory concern questions such as what one can know about a quantum system and how the information encoded therein can be processed and distributed between parties. Another challenging feature of quantum theory is its incompatibility with the other cornerstone of modern physics, general relativity. In order to reconcile the two, one must identify and retain only the essential concepts and principles of each theory, and in the case of general relativity, causality has been identified as such a concept. This raises the question of how our classical understanding of causality must change when quantum theory is taken into account. In order to address these questions about information, knowledge and causality, I turn to the framework of causal models. In classical statistics, causal models explain the relations among a set of variables in terms of causal influences, which makes them a powerful tool for structuring our knowledge about complex systems and developing strategies for interacting with them. More importantly, the framework provides the conceptual underpinnings and mathematical methods for addressing questions about causation, information and knowledge in a rigorous manner. In this thesis, I develop a version of the classical causal models framework that is compatible with quantum theory and explore its physical implications. \n \n \nA first step is to define quantum versions of fundamental elements such as variables, conditionals and belief propagation rules. This allows one to consider the question of what one can come to know about a quantum variable from the point of view of causal modelling. The conditionals relating quantum variables are found to have a richer structure than their classical counterparts, which can be exploited for the task of discerning causal relations given limited data -- a central problem in classical causal modelling. The mathematical properties of quantum conditionals also establish a correspondence between various classes of two-party correlations, such as bound and distillable entanglement, and the types of causal structures that can give rise to them, which may become a useful tool for entanglement theory and quantum information processing. In the context of open quantum systems dynamics, quantum causal models provide a clear physical explanation of not completely positive maps and, more broadly, of non-Markovian quantum dynamics. Finally, I consider the possibility of non-classical effects in the way that different causal mechanisms are combined -- that is, non-classical causal structures. For the simple case of two causally ordered variables, I propose indicators that witness different classes of combinations of causal mechanisms, including a non-classical mixture, and describe an experiment realizing examples of the different classes. \n \n \nAs these results illustrate, the framework of quantum causal models provides both greater conceptual clarity and a comprehensive mathematical formalism for studying a diverse set of problems, ranging from foundational questions to applications in open systems dynamics and the exploration of non-classical causal structures, which are likely to feature in a future theory of quantum gravity. The material presented in this thesis is intended as a foundation and inspiration for further applications of quantum causal models.
Causalism is the traditional attitude of disowning all noncausal categories of determination, holding dogmatically that every connection in the world is causal. The entirely negative attitude taken by indeterminism and by empiricism toward the principle of causation is inconsistent with the very goal of science, which is the search for the objective forms of determination and interconnection. This chapter deals with the question concerning the range of validity of the causal principle. The notion of causal range of the laws of nature will be clarified and illustrated, by analyzing in some detail a typical physical law with which every student of electricity is familiar. Science seems to show neither an enlargement nor a progressive shrinking of the domain of validity of the causal principle. Most scientists and philosophers of science believed that quantum mechanics had given causality the death blow, by showing that quantum phenomena are inherently random, hence only statistically predictable.
Quantum theory is the soul of contemporary physics. It was discovered in an adventurous way, under the urge to solve the puzzles posed by atomic spectra and blackbody radiation. But after its invention, it immediately became clear that it was not just a theory of specific physical systems: it was rather a new language of universal applicability. Already in 1928, the theory had received solid mathematical foundations by Hilbert, von Neumann, and Nordheim, and this work was brought to completion in the monumental work of von Neumann, in the form that we still study nowadays. The theory is extraordinarily successful, and its predictions have been confirmed to an astonishing level of precision in a large spectrum of experiments. However, almost 90 years after von Neumann's book, quantum theory remains mysterious. Its mathematical formulation – based on Hilbert spaces and self-adjoint operators – is far from having an intuitive interpretation. The association of physical systems to Hilbert spaces whose unit vectors represent pure states, the representation of transformations by unitary operators and of observables by self-adjoint operators – all such postulates look artificial and ad hoc . A slightly more operational approach is provided by the C * -algebraic formulation of quantum theory – still, this formulation relies on the assumption that observables form an algebra , where the physical meanings of the multiplication and the sum are far from clear. In short, the postulates of quantum theory impose mathematical structures without providing any simple reason for this choice: the mathematics of Hilbert spaces is adopted as a magic blackbox that “works well” at producing experimental predictions. However, in a satisfactory axiomatization of a physical theory the mathematical structures should emerge as a consequence of postulates that have a direct physical interpretation. By this we mean postulates referring, e.g., to primitive notions like physical system, measurement, or process, rather than notions like, e.g., Hilbert space, C * -algebra, unit vector, or selfadjoint operator. The crucial question thus remains unanswered: why quantum theory? Which are the principles at the basis of the theory? A case that is often invoked in contrast is that of Special Relativity theory, which directly follows from the simple understandable principle of relativity.
A key aim of quantum foundations is to characterize the sense in which nature goes beyond classical physics. Understanding nonclassicality is one of our best avenues towards finding a satisfactory interpretation of quantum theory. By determining which classical principles cannot be satisfied in any empirically adequate physical theory, we begin to see which principles can be preserved, which in turn gives us insight into the ontology of the world. These insights then guide us in determining which questions to ask and which experiments to perform next. Furthermore, it is these nonclassical aspects of nature that give rise to new technologies such as the speed-ups of quantum computation or the security of quantum key distribution. \n \nThe gold standard for establishing that a phenomenon is truly nonclassical is to prove that it violates the principle of local causality or the principle of noncontextuality. Much of this thesis reports on my research relating to these two principles. This research primarily involves (i) finding new justifications for our notions of nonclassicality; (ii) refining their fundamental definitions; (iii) quantifying and characterizing their various manifestations; and (iv) finding applications where nonclassical phenomena act as resources for information processing. \n \nUltimately, all of this work on nonclassicality is woven together into a novel framework for physical theories introduced by myself, John Selby, and Rob Spekkens. Its main advantage over preexisting frameworks is that it maintains a clear distinction between which elements of a given physical theory directly describe causal processes, and which refer only to one’s inferences about causal processes. This clarifies a number of confusions in the literature which arose precisely because previous frameworks scrambled causal and inferential concepts. Furthermore, local causality and noncontextuality emerge in this framework as the assumptions that the causal and inferential structures (respectively) that are operationally observed must be respected in the underlying ontology. This work constitutes a first step in developing a new interpretation of quantum theory—the first interpretation designed to satisfy the spirit of both local causality and noncontextuality.