Phase plays a crucial role in many quantum effects including interference. Here we lay the foundations for the study of phase in probabilistic theories more generally. Phase is normally defined in terms of complex numbers that appear when representing quantum states as complex vectors. Here we give an operational definition whereby phase is instead defined in terms of measurement statistics. Our definition is phrased in terms of the operational framework known as generalized probabilistic theories or the convex framework . The definition makes it possible to ask whether other theories in this framework can also have phase. We apply our definition to investigate phase and interference in several example theories: classical probability theory, a version of Spekkens' toy model, quantum theory and box-world. We find that phase is ubiquitous; any non-classical theory can be said to have non-trivial phase dynamics.
Phase plays a crucial role in many quantum effects including interference. Here we lay the foundations for the study of phase in probabilistic theories more generally. Phase is normally defined in terms of complex numbers that appear when representing quantum states as complex vectors. Here we give an operational definition whereby phase is instead defined in terms of measurement statistics. Our definition is phrased in terms of the operational framework known as generalized probabilistic theories or the convex framework . The definition makes it possible to ask whether other theories in this framework can also have phase. We apply our definition to investigate phase and interference in several example theories: classical probability theory, a version of Spekkens' toy model, quantum theory and box-world. We find that phase is ubiquitous; any non-classical theory can be said to have non-trivial phase dynamics.
We give an introduction to research associated with the generalised probabilistic theories framework, also known as the convex framework. States are real vectors representing lists of probabilities of measurement outcomes. Convex combinations of the vectors represent probabilistic combinations of different state preparations. Transformations are real matrices. Measurement outcomes are represented by functionals of the states, inner products of the state with a real vector, whose values are the probability of the measurement outcome in question. The framework generalises quantum theory. We describe the operational meaning of the framework, and how the concepts can be defined in terms of cones of states and measurement outcome vectors. We describe how the classical and quantum probability theories are represented in the framework. We describe Bell non-locality and the theory with super-quantum non-locality known as box world. We discuss generalised Hamiltonian mechanics in the discrete case and in continuous phase space, including the role of negativity of the phase space density in contextuality and tunnelling.
Phase lies at the heart of quantum physics and quantum information theory. A quantum bit is qualitatively different from a classical bit as it allows for the coherent superposition of possibilities, which demonstrate different behaviours depending on the phase between them. These behaviours constitute as interference phenomena, and lie behind the existence of algorithms in quantum computing which are arguably faster than the best classical alternatives. The concept of phase is deeply steeped in the structure of Hilbert spaces: the mathematical framework that underlies quantum theory. What if quantum theory did not hold in all scenarios, or was only a limiting case of some broader theory? In this case, would we still be able to meaningfully talk about phase and interference? In this thesis, we will adopt an operational generalisation of quantum theory known as the framework of generalised probabilistic theories. We will provide a reasonable definition of phase in this framework. Using this, we shall explore single-particle interferometry set-ups (particularly Mach-Zehnder interferometers): experiments whose output is highly dependent on the phase between the spatially disjoint branches through which a particle might be traversing. By applying physically-motivated locality considerations, we identify the crucial role that the uncertainty principle and its generalisations play in quantum theory as an enabler of non-trivial interference. By taking into account relativity of simultaneity, we will also provide a physical motivation for why standard quantum theory provides the best description of the location of a particle traversing such a system. Finally, we apply our generalised definition of phase in the related context of particle exchange behaviour, and identify a method for classifying post-quantum particles. All of this will demonstrate that phase between possibilities and its consequences are not uniquely quantum phenomena. Much of the behaviour we might ascribe to phase in quantum theory in fact holds generally true for phase in probabilistic theories.
The framework of generalized probabilistic theories is a popular approach for studying the physical foundations of quantum theory. The standard framework assumes the no-restriction hypothesis, in which the state space of a physical theory determines the set of measurements. However, this assumption is not physically motivated. We generalize the framework to account for systems that do not obey the no-restriction hypothesis. We then show how our framework can be used to describe certain classes of probabilistic theories, for example, those which include intrinsic noise. Relaxing the restriction hypothesis also allows us to introduce a ``self-dualization'' procedure, which yields a class of theories that share many features of quantum theory. We then characterize joint states, generalizing the maximal tensor product. We show how this tensor product can be used to describe the convex closure of the Spekkens toy theory, and in doing so we obtain an analysis of why it is local in terms of the geometry of its state space. We show that the unrestricted version of the Spekkens toy theory is the theory known as ``boxworld'' that allows maximal nonlocal correlations.
The law of addition of probabilities by means of complex probability amplitudes with its powerful interference consequences, unique to quantum mechanics, is shown to apply elsewhere. The two parts of quantum mechanics: an unvisualizable dynamics and a visualizable concretization (the latter corresponding to reduction of the state vector when a measurement is made) are shown to have a correspondence in computation. Namely, a conceptual structure called a field (of the real numbers, say), its operations, and the concretization of that structure is done by means of a (digital) computer (the latter is interpreted as playing the role of a measuring apparatus). The probabilistic state-reduction operator of quantum mechanics is replaced by a deterministic operation, an extension of rounding. State, wave function, dynamics, observation, uncertainty, and nonlocality are shown to have their counterparts in the new model. The algorithmic counterpart of the double slit experiment to validate the existence of interference in the new framework is defined and performed. A notion of spin in computation is introduced. In an appendix, we comment on how this model impacts a speculative theory of the mind by Penrose.
This thesis explores various perspectives on quantum phenomena, and how our understanding of these phenomena is informed by the study of general probabilistic theories. Particular attention is given to quantum nonlocality, and its interaction with areas of physical and mathematical interest such as entropy, reversible dynamics, information-based games and the idea of negative probability. We begin with a review of non-signaling distributions and convex operational theories, including “black box” descriptions of experiments and the mathematics of convex vector spaces. In Chapter 3 we derive various classical and quantum-like quasiprobabilistic representations of arbitrary non-signaling distributions. Previously, results in which the density operator is allowed to become non-positive [1] have proved useful in derivations of quantum theory from physical requirements [2]; we derive a dual result in which the measurement operators instead are allowed to become non-positive, and show that the generation of any non-signaling distribution is possible using a fixed separable state with negligible correlation. We also derive two distinct “quasi-local” models of non-signaling correlations. Chapter 4 investigates non-local games, in particular the game known as Information Causality. By analysing the probability of success in this game, we prove the conjectured tightness of a bound given in [3] concerning how well entanglement allows us to perform the task of random access coding, and introduce a quadratic bias bound which seems to capture a great deal of information about the set of quantum-achievable correlations. By reformulating Information Causality in terms of entropies, we find that a sensible measure of entropy precludes many general probabilistic theories whose non-locality is stronger than that of quantum theory. Chapter 5 explores the role that reversible transitivity (the principle that any two pure states are joined by a reversible transformation) plays as a characteristic feature of quantum theory. It has previously been shown that in Boxworld, the theory allowing for the full set of non-signaling correlations, any reversible transformation on a restricted class of composite systems is merely a composition of relabellings of measurement choices and outcomes, and permutations of subsystems [4]. We develop a tabular description of Boxworld states and effects first introduced in [5], and use this to extend this reversibility result to any composite Boxworld system in which none of the subsystems are classical.
In this paper we propose a unified conceptual framework in which quantum probability, quantum logic, and classical probability theory are understood as different structural levels of a single theoretical hierarchy. Starting from a statistical interpretation of quantum interference, previously developed by the authors, we argue that quantum probability is not an alternative to classical Kolmogorov probability, but a more general structure. Classical probability theory emerges as a limiting case corresponding to the disappearance of interference and the effective decomposition of the whole into independent elements. Within this framework, the Birkhoff–von Neumann quantum logic appears as a natural logical expression of quantum probability rather than as an abstract algebraic construction detached from physical meaning. The priority of the whole over its parts plays a central role in both quantum probability and quantum logic, while classical logic and probability arise when this priority becomes trivial. We emphasize that the proposed interpretation does not reject existing quantum formalism, but reinterprets its logical and probabilistic foundations. This approach may have implications for the conceptual understanding of quantum algorithms and quantum information processing.