There has been a body of work deriving the complex Hilbert-space structure of quantum theory from axioms/principles/postulates to deepen our understanding of quantum theory and to reveal ways to go beyond it to resolve foundational issues. Recent progress in incorporating indefinite causal structure into physical theories suggests that a more comprehensive understanding of both quantum theory and the theory beyond it accounts for indefinite causal structure. We formulate a framework of physical theories without assuming definite causal structure and identify postulates that single out the complex Hilbert-space structure. More than one complex Hilbert-space theory is compatible with the postulates, which leaves room for further search for the best among these theories.
There has been a body of work deriving the complex Hilbert-space structure of quantum theory from axioms/principles/postulates to deepen our understanding of quantum theory and to reveal ways to go beyond it to resolve foundational issues. Recent progress in incorporating indefinite causal structure into physical theories suggests that a more comprehensive understanding of both quantum theory and the theory beyond it accounts for indefinite causal structure. We formulate a framework of physical theories without assuming definite causal structure and identify postulates that single out the complex Hilbert-space structure. More than one complex Hilbert-space theory is compatible with the postulates, which leaves room for further search for the best among these theories.
It is not surprising that our language should be incapable of describing processes occurring within atoms, for, as has been remarked, it was invented to describe the experiences of daily life, … Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme–the quantum theory–which seems entirely adequate for the treatment of atomic processes. Werner Heisenberg, The Physical Principles of the Quantum Theory (1930) In this chapter we introduce the mathematical structure of quantum theory, for a finite number of systems and finite dimension, starting from three simple Hilbert-space postulates, and based on the intuitive notions of state, transformation, and effect, within a causal context. Such notions will be formalized in a general non-causal context in Chapter 3, which is devoted to the informational framework. We will assume the reader has a basic knowledge of matrix analysis and of operators over Hilbert spaces. Comments about the extension to infinite dimensions, continuous outcome-spaces, and infinitely many systems can be found in the notes at the end of the chapter. From the three simple Hilbert-space postulates, the present chapter extracts six principles, which will become the postulates for quantum theory in the axiomatic derivation presented in Part III of the book. In addition, we will take the opportunity to present and explore many relevant features of the theory – including its convex and causal structure, the Choi–Jamiołkowski isomorphism, and relevant notions, such as those of POVM, complete positivity of maps, quantum operation and channels, entangled states, state discrimination, tomography of states, and transformations. The purification principle will open the stage to the dilation theorems of the theory of quantum open systems, along with other consequences, such as the existence of faithful and steering states, the theorem of no information without disturbance, quantum teleportation, and more. We will review the quantum no-cloning and no-programming theorems, and finally provide a thorough derivation of the theory of quantum channel inversion, which is the backbone of the general theory of quantum error correction in quantum information. Additional interesting topics will be classical information theory as a restriction of quantum information theory, and a critical reconsideration of the von Neumann postulate and the notion of observable.
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.
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.
We adopt an operational mode of thinking to study spacetime fluctuations and their impacts on questions of quantum gravity. Particular emphasis is put on studying causal structure fluctuations. The study is guided by the principle of causal neutrality, which says that fundamental concepts and laws of physics should be stated without assuming a definite spacetime causal structure. Most traditional concepts and theories of physics violate this principle. A major theme of this thesis is to upgrade the traditional concepts and theories in accordance with the principle of causal neutrality. \n \nAmong other things, the thesis include works on the following. (1) An axiomatic derivation of the complex Hilbert space structure of quantum theory without assuming definite causal structure. (2) A so-called causally neutral quantum field theory (CNQFT) framework for algebraic quantum physics allowing indefinite causal structure. (3) A proposal that causal fluctuation regularizes quantum field ultraviolet divergences (UV) and that the UV regularizing correlation functions with causal fluctuations characterize the UV structure of physical states. (4) A study on quantitative measures of causality. (5) A so-called ``correlation networks'' framework to study quantum theory with indefinite causal structure, which generalizes previous frameworks. (6) New definitions of entanglement and entanglement measures that conform to the causal neutrality principle. (7) Ideas on finding a causally neutral analogue of Einstein's equation for quantum spacetime. (8) A simple theorem showing that the communication capacities of general correlations (with definite or indefinite causal structure) defined with respect to state transmission can be reduced to those of channels.
The Causaloid framework is an operational approach aimed to house both the radical aspects of General Relativity -- dynamic causal structure, and Quantum Theory -- indefiniteness, to provide a scaffolding that might be suitable for Quantum Gravity by providing a landscape of theories that allow for indefinite causal structure. One may consider it as a generalisation of generalised probability theories (or GPTs) where a priori regions are not assumed to have any given causal relationship, to incorporate the possibility of indefinite causal structure. Since its conception, there have been many advances in the field of indefinite causal structure mostly stemming from the work of Chiribella et al. on the quantum switch and supermaps and from Oreshkov et al. on causal inequalities and process matrices. These approaches have systems moving along wires and use Hilbert space structure. They violate the standard causality constraints of Quantum Theory and, in this sense, can be regarded as post-quantum. The Causaloid approach does not necessarily have systems moving along wires or Hilbert spaces. This is the first paper in a trilogy of papers aiming to close the gap between the Causaloid (that allows for GPTs) and post-quantum studies that employ Hilbert spaces. To do so in the present paper, we provide a diagrammatic language for the Causaloid framework along with new terminology for the three levels of physical compression (called Tomographic, Compositional, and Meta compression).
‘Quantum theory’ is not a single physical theory but a framework in which many different concrete theories fit. As such, a solution to the quantum measurement problem ought to provide a recipe to interpret each such concrete theory, in a mutually consistent way. But with the exception of the Everett interpretation, the main extant solutions either try to make sense of the abstract framework as if it were concrete, or else interpret one particular quantum theory under the fiction that it is fundamental and exact. In either case, these approaches are unable to help themselves to the very theory-laden, level-relative ways in which quantum theory makes contact with experiment in mainstream physics, and so are committed to major revisionary projects which have not been carried out even in outline. As such, only the Everett interpretation is currently suited to make sense of quantum physics as it is found.