Related papers: An effect-theoretic reconstruction of quantum theo…
The idea that symmetries simplify or reduce the complexity of a system has been remarkably fruitful in physics, and especially in quantum mechanics. On a mathematical level, symmetry groups single out a certain structure in the Hilbert…
We first show that the convex effect algebras (CEA) approach to quantum mechanics is more general than the general probabilistic theories approach. We then restrict our attention to finite-dimension CEA's. After an introductory Section~1,…
Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serves as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of…
This paper is the first of several parts introducing a new powerful algebra: the algebra of the pseudo-observables. This is a C*-algebra whose set is formed by formal expressions involving observables. The algebra is constructed by applying…
C*-algebras are widely used in mathematical physics to represent the observables of physical systems, and are sometimes taken as the starting point for rigorous formulations of quantum mechanics and classical statistical mechanics.…
The aim of this article is to describe a class of *-algebras that allows to treat well-behaved algebras of unbounded operators independently of a representation. To this end, Archimedean ordered *-algebras (*-algebras whose real linear…
This survey article is concerned with the modeling of the kinematical structure of quantum systems in an algebraic framework which eliminates certain conceptual and computational difficulties of the conventional approaches. Relying on the…
Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras E we investigate a natural implication and prove that the implication reduct of E is term equivalent to E. Then we present a…
The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr's idea that the empirical content of quantum physics is accessible…
We provide a reformulation of finite dimensional quantum theory in the circuit framework in terms of mathematical axioms, and a reconstruction of quantum theory from operational postulates. The mathematical axioms for quantum theory are the…
The purely algebraic notion of CQG algebra (algebra of functions on a compact quantum group) is defined. In a straightforward algebraic manner, the Peter-Weyl theorem for CQG algebras and the existence of a unique positive definite Haar…
We reconstruct finite-dimensional quantum theory from categorical principles. That is, we provide properties ensuring that a given physical theory described by a dagger compact category in which one may `discard' objects is equivalent to a…
It is a fundamental prediction of quantum theory that states of physical systems are described by complex vectors or density operators on a Hilbert space. However, many experiments admit effective descriptions in terms of other state…
The Jordan structure of finite-dimensional quantum theory is derived, in a conspicuously easy way, from a few simple postulates concerning abstract probabilistic models (each defined by a set of basic measurements and a convex set of…
Quantum and classical mechanics share a common algebraic formalism which is expressed naturally in the language of category theory. A third realization of this formalism is the so-called hyperbolic quantum mechanics where split-complex…
Carrying the insights of conditional probability to the quantum realm is notoriously difficult due to the non-commutative nature of quantum observables. Nevertheless, conditional expectations on von Neumann algebras have played a…
Formulations of quantum mechanics can be characterized as realistic, operationalist, or a combination of the two. In this paper a realistic theory is defined as describing a closed system entirely by means of entities and concepts…
Qubits are a great way to build a quantum computer, but a limited way to program one. We replace the usual "states and gates" formalism with a "props and ops" (propositions and operators) model in which (a) the C*-algebra of observables…
We study non-selfadjoint operator algebras that can be entirely understood via their finite-dimensional representations. In contrast with the elementary matricial description of finite-dimensional $\mathrm{C}^*$-algebras, in the…
Decompositional theories describe the ways in which a global physical system can be split into subsystems, facilitating the study of how different possible partitions of a same system interplay, e.g. in terms of inclusions or signalling. In…