In this chapter we consider two examples of the situation when the classicalobservables should be described by a noncommutative (quantum-like)probability space. A possible experimental approach to find quantum-like correlationsfor classical disordered systems is discussed. The interpretation ofnoncommutative probability in experiments with classical systems as a resultof context (complex of experimental physical conditions) dependence ofprobability is considered.
In this chapter we consider two examples of the situation when the classicalobservables should be described by a noncommutative (quantum-like)probability space. A possible experimental approach to find quantum-like correlationsfor classical disordered systems is discussed. The interpretation ofnoncommutative probability in experiments with classical systems as a resultof context (complex of experimental physical conditions) dependence ofprobability is considered.
We study the role of context, complex of physical conditions, in quantum as well as classical experiments. It is shown that by taking into account contextual dependence of experimental probabilities we can derive the quantum rule for the addition of probabilities of alternatives. Thus we obtain quantum interference without applying to wave or Hilbert space approach. The Hilbert space representation of contextual probabilities is obtained as a consequence of the elementary geometric fact: cos-theorem. By using another fact from elementary algebra we obtain complex-amplitude representation of probabilities. Finally, we found contextual origin of noncommutativity of incompatible observables.
We study the role of context, complex of physical conditions, in quantum as well as classical experiments. It is shown that by taking into account contextual dependence of experimental probabilities we can derive the quantum rule for the addition of probabilities of alternatives. Thus we obtain quantum interference without applying the wave or Hilbert space approach. The Hilbert space representation of contextual probabilities is obtained as a consequence of the elementary geometric fact: cos-theorem. By using another fact from elementary algebra we obtain complex-amplitude representation of probabilities. Finally, we found contextual origin of noncommutativity of incompatible observables.
Abstract: A suitable unified statistical formulation of quantum and classical mechanics in a ∗-algebraic setting leads us to conclude that information itself is noncommutative in quantum mechanics. Specifically we refer here to an observer’s information regarding a physical system. This is seen as the main difference from classical mechanics, where an observer’s information regarding a physical system obeys classical probability theory. Quantum mechanics is then viewed purely as a mathematical framework for the probabilistic description of noncommutative information, with the projection postulate being a noncommutative generalization of conditional probability. This view clarifies many problems surrounding the interpretation of quantum mechanics, particularly problems relating to the measuring process. 1.
This chapter introduces the mathematical framework, basic rules, and some key results of quantum theory. After a succinct overview of linear algebra and an introduction to complex Hilbert space, it investigates the correspondence between subspaces of Hilbert space and propositions, their logical structure, and how the pertinent probabilities are calculated. It discusses the mathematical representation of states, observables, and transformations, as well as the rules for calculating expectation values and uncertainties, and for updating states after a measurement. Particular attention is paid to two-level systems, or ‘qubits’, and the connection is made with experimental evidence about binary measurements. The properties of composite systems are discussed in detail, notably the phenomenon of entanglement. The chapter concludes with an investigation of conceptual issues regarding realism, non-contextuality, and locality, as well as the classical limit.
We continue the development of a so-called contextual statistical model (here context has the meaning of a complex of physical conditions). It is shown that, besides contexts producing the conventional trigonometric cos-interference, there exist contexts producing the hyperbolic cos-interference. Starting with the corresponding interference formula of total probability we represent such contexts by hyperbolic probabilistic amplitudes or in the abstract formalism by normalized vectors of a hyperbolic analogue of the Hilbert space. There is obtained a hyperbolic Born’s rule. Incompatible observables are represented by noncommutative operators. This paper can be considered as the first step towards hyperbolic quantum probability. We also discuss possibilities of experimental verification of hyperbolic quantum mechanics: in physics of elementary particles, string theory as well as in experiments with nonphysical systems, e.g., in psychology, cognitive sciences, and economy.
The origin and basis of the notion of quantum contextuality is identified in the Copenhagen approach to quantum mechanics, where context is automatically invoked by its requirement that the experimental arrangement involved in any measurements or set of measurements be taken into account while, in general, the outcome of a measurement may depend on other measurements immediately preceding or jointly performed on the same system. For Bohr, the specification of the experimental situation of any measurement is essential to its significance in light of complementarity and the omnipresence of the quantum of action in physics; for Heisenberg, the incompatibility of pairs of sharp measurements belonging to different situations coheres with both the completeness of the quantum state as an objective physical description and the principle of indeterminacy. Here, context in the Copenhagen approach is taken to be the equivalence class of experimental arrangements corresponding to a set of compatible measurements of quantum observables in standard quantum mechanics; the associated form of contextuality in quantum mechanics arises via the non-commutativity in general of sharp observables, proven by von Neumann, that can appear, providing different contexts. This notion is related to theoretical situations explored later by Bell, by Kochen and Specker, and by others in relation to the classification of hidden-variables theories and elsewhere in physics. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.