Generalized Probabilistic Theories in a New Light

In this paper we will present a modified formulation of generalized probabilistic theories that will always give rise to the structure of Hilbert space of quantum mechanics, in any finite outcome space, and give the guidelines to how to extend this work to infinite dimensional Hilbert spaces. Moreover, this new formulation which we will call extended operational-probabilistic theories, applies not only to quantum systems, but also equally well to classical systems, without violating Bell’s theorem, and at the same time solves the measurement problem. This is why we will see that the question of why our universe is quantum mechanical rather than classical is misplaced. The only difference that exists between a classical universe and a quantum mechanical one lies merely in which observables are compatible and which are not. Besides, this extended probability theory which we present in this paper shows that it is non-determinacy, or to be more precise, the non-deterministic description of the universe, that makes the laws of physics the way they are. In addition, this paper shows us that what used to be considered as purely classical systems and to be treated that way are in fact able to be manipulated according to the rules of quantum mechanics –with this new understanding of these rules- and that there is still a possibility that there might be a deterministic level from which our universe emerges, which if understood correctly, may open the door wide to applications in areas such as quantum computing. In addition to all that, this paper shows that without the use of complex vector spaces, we cannot have any kind of continuous evolution of the states of any system.

Probabilistic theories and reconstructions of quantum theory

These lecture notes provide a basic introduction to the framework of generalized probabilistic theories (GPTs) and a sketch of a reconstruction of quantum theory (QT) from simple operational principles. To build some intuition for how physics could be even more general than quantum, I present two conceivable phenomena beyond QT: superstrong nonlocality and higher-order interference. Then I introduce the framework of GPTs, generalizing both quantum and classical probability theory. Finally, I summarize a reconstruction of QT from the principles of Tomographic Locality, Continuous Reversibility, and the Subspace Axiom. In particular, I show why a quantum bit is described by a Bloch ball, why it is three-dimensional, and how one obtains the complex numbers and operators of the usual representation of QT.

The Operational Choi–Jamiołkowski Isomorphism

In this article, I use an operational formulation of the Choi–Jamiołkowski isomorphism to explore an approach to quantum mechanics in which the state is not the fundamental object. I first situate this project in the context of generalized probabilistic theories and argue that this framework may be understood as a means of drawing conclusions about the intratheoretic causal structure of quantum mechanics which are independent of any specific ontological picture. I then give an operational formulation of the Choi–Jamiołkowski isomorphism and show that, in an operational theory which exhibits this isomorphism, several features of the theory which are usually regarded as properties of the quantum state can be derived from constraints on non-local correlations. This demonstrates that there is no need to postulate states to be the bearers of these properties, since they can be understood as consequences of a fundamental equivalence between multipartite and temporal correlations.

An effect-theoretic reconstruction of quantum theory

An often used model for quantum theory is to associate to every physical system a C∗-algebra. From a physical point of view it is unclear why operator algebras would form a good description of nature. In this paper, we find a set of physically meaningful assumptions such that any physical theory satisfying these assumptions must embed into the category of finite-dimensional C∗-algebras. These assumptions were originally introduced in the setting of effectus theory, a categorical logical framework generalizing classical and quantum logic. As these assumptions have a physical interpretation, this motivates the usage of operator algebras as a model for quantum theory.In contrast to other reconstructions of quantum theory, we do not start with the framework of generalized probabilistic theories and instead use effect theories where no convex structure and no tensor product needs to be present. The lack of this structure in effectus theory has led to a different notion of pure maps. A map in an effectus is pure when it is a composition of a compression and a filter. These maps satisfy particular universal properties and respectively correspond to `forgetting' and `measuring' the validity of an effect.We define a pure effect theory (PET) to be an effect theory where the pure maps form a dagger-category and filters and compressions are adjoint. We show that any convex finite-dimensional PET must embed into the category of Euclidean Jordan algebras. Moreover, if the PET also has monoidal structure, then we show that it must embed into either the category of real or complex C∗-algebras, which completes our reconstruction.