In this paper we derive the complex Hilbert space formalism of quantum theory from four simple information theoretic axioms. It is shown that quantum theory is the only non classical probabilistic theory satisfying the following axioms: distinguishability, conservation, reversibility, composition. The new results of this reconstruction compared to other reconstructions by other authors are: (i) we get rid of axiom "subspace" in favor of axiom conservation eliminating mathematical requirements contained in previous axiomatics; (ii) we are able to classify all the probabilistic theories that are consistent requiring (a) only the first two axioms (b) only the first three axioms; this could be useful in experimental tests of quantum theory since it gives the possibility to understand whether or not other mathematical models could be consistent with such tests; (iii) we provide a connection between two different approaches to quantum foundations, quantum logic and the one based on information theoretic primitives showing that any theory satisfying the first two axioms given above either is classical or is a theory in which physical systems are described by a projective geometry.
In this paper we derive the complex Hilbert space formalism of quantum theory from four simple information theoretic axioms. It is shown that quantum theory is the only non classical probabilistic theory satisfying the following axioms: distinguishability, conservation, reversibility, composition. The new results of this reconstruction compared to other reconstructions by other authors are: (i) we get rid of axiom "subspace" in favor of axiom conservation eliminating mathematical requirements contained in previous axiomatics; (ii) we are able to classify all the probabilistic theories that are consistent requiring (a) only the first two axioms (b) only the first three axioms; this could be useful in experimental tests of quantum theory since it gives the possibility to understand whether or not other mathematical models could be consistent with such tests; (iii) we provide a connection between two different approaches to quantum foundations, quantum logic and the one based on information theoretic primitives showing that any theory satisfying the first two axioms given above either is classical or is a theory in which physical systems are described by a projective geometry.
In this paper we derive the complex Hilbert space formalism of quantum theory from four simple information theoretic axioms. It is shown that quantum theory is the only non classical probabilistic theory satisfying the following axioms: distinguishability, conservation, reversibility, composition. The new results of this reconstruction compared to other reconstructions by other authors are: (i) we get rid of axiom subspace in favor of axiom conservation eliminating mathematical requirements contained in previous axiomatics; (ii) we are able to classify all the probabilistic theories that are consistent requiring (a) only the first two axioms (b) only the first three axioms; this could be useful in experimental tests of quantum theory since it gives the possibility to understand whether or not other mathematical models could be consistent with such tests; (iii) we provide a connection between two different approaches to quantum foundations, quantum logic and the one based on information theoretic primitives showing that any theory satisfying the first two axioms given above either is classical or is a theory in which physical systems are described by a projective geometry.
We review a recent approach to the foundations of quantum mechanics inspired by quantum information theory [1, 2]. The approach is based on a general framework, which allows one to address a large class of physical theories which share basic information-theoretic features. We first illustrate two very primitive features, expressed by the axioms of causality and purity-preservation, which are satisfied by both classical and quantum theory. We then discuss the axiom of purification, which expresses a strong version of the Conservation of Information and captures the core of a vast number of protocols in quantum information. Purification is a highly non-classical feature and leads directly to the emergence of entanglement at the purely conceptual level, without any reference to the superposition principle. Supplemented by a few additional requirements, satisfied by classical and quantum theory, it provides a complete axiomatic characterization of quantum theory for finite dimensional systems.
We review a recent approach to the foundations of quantum mechanics inspired by quantum information theory [1, 2]. The approach is based on a general framework, which allows one to address a large class of physical theories which share basic information-theoretic features. We first illustrate two very primitive features, expressed by the axioms of causality and purity-preservation, which are satisfied by both classical and quantum theory. We then discuss the axiom of purification, which expresses a strong version of the Conservation of Information and captures the core of a vast number of protocols in quantum information. Purification is a highly non-classical feature and leads directly to the emergence of entanglement at the purely conceptual level, without any reference to the superposition principle. Supplemented by a few additional requirements, satisfied by classical and quantum theory, it provides a complete axiomatic characterization of quantum theory for finite dimensional systems.
We propose a system of information-theoretic axioms from which we derive the formalism of quantum theory. Part I is devoted to the conceptual foundations of the information-theoretic approach. We argue that this approach belongs to the epistemological framework depicted as a loop of existences. In Part II we derive the quantum formalism from information-theoretic axioms and we analyze the twofold role of observer as physical system and as informational agent. Quantum logical techniques are then introduced, and we prove theorems reconstructing elements of the formalism. In Part III, we introduce the formalism of C*-algebras and give it an information theoretic interpretation. We analyze the conceptual underpinnings of the theory of modular automorphisms and we give an information-theoretic justification for the emergence of time in the algebraic approach. We conclude by giving a list of open questions, including topics in cognitive science, decision theory, and information technology.
Abstract The reconstruction of quantum physics has been connected with the interpretation of the quantum formalism, and has continued to be so with the recent deeper consideration of the relation of information to quantum states and processes. This recent form of reconstruction has mainly involved conceiving quantum theory on the basis of informational principles, providing new perspectives on physical correlations and entanglement that can be used to encode information. By contrast to the traditional, interpretational approach to the foundations of quantum mechanics, which attempts directly to establish the meaning of the elements of the theory and often touches on metaphysical issues, the newer, more purely reconstructive approach sometimes defers this task, focusing instead on the mathematical derivation of the theoretical apparatus from simple principles or axioms. In its most pure form, this sort of theory reconstruction is fundamentally the mathematical derivation of the elements of theory from explicitly presented, often operational principles involving a minimum of extra‐mathematical content. Here, a representative series of specifically information‐based treatments—from partial reconstructions that make connections with information to rigorous axiomatizations, including those involving the theories of generalized probability and abstract systems—is reviewed.
This chapter provides a probabilistic framework for formulating classical probability theory, quantum probability, thermodynamics, diffusion, and the Wiener integral using a set of four axioms or principles. It explains everything that conventional quantum information theory and classical probability theory achieve. We want to emphasize that this framework is not an interpretation of quantum mechanics such as “Many-Worlds,” “Bohm’s Theory,” or the “Copenhagen interpretation.” It is much more general and can be viewed as a probability algorithm that calculates probabilities of future events. As a result, previously perplexing paradoxes find resolution. In particular, the superposition principle takes on a new meaning. Our probabilistic framework stands apart from the Hilbert space formalism. It relies solely on elementary set theory, classical logic, and complex numbers. Consequently, this theory is accessible for instruction in educational settings. This framework can be regarded as an axiomatic approach to probability in the sense of Hilbert. In his sixth of the twenty-three open problems presented at the International Congress of Mathematicians in Paris in 1900, Hilbert called for an axiomatic probability theory.