From quantum probabilities to classical facts (original) (raw)
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On classical systems and measurements in quantum mechanics
Quantum Studies: Mathematics and Foundations, 2019
The recent rigorous derivation of the Born rule from the dynamical law of quantum mechanics [Phys. Rep. 525(2013) 1-166] is taken as incentive to reexamine whether quantum mechanics has to be an inherently probabilistic theory.
Mathematics
The link between classical and quantum theories is discussed in terms of extensional and intensional viewpoints. The paper aims to bring evidence that classical and quantum probabilities are related by intensionalization, which means that by abandoning sets from classical probability one should obtain quantum theory. Unlike the extensional concept of a set, the intensional probability is attributed to the quantum ensemble, which is contextually dependent. The contextuality offers a consistent realization of the measurement problem, which should require the existence of the time operator. The time continuum by Brouwer has satisfied such a requirement, which makes it fundamental to mathematical physics. The statistical model it provides has been proven tremendously useful in a variety of applications.
Elements of Quantum Probability
This is an introductory article presenting some basic ideas of quantum probability. From a discussion of simple experiments with polarized light and a card game we deduce the necessity of extending the body of classical probability theory. For a class of systems, containing classical systems with finitely many states, a probabilistic model is developed. It can describe, in particular, the polarization experiments. Some examples of quantum coin tosses are discussed, closely related to V.F.R. Jones approach to braid group representations, to spin relaxion, and to nuclear magnetic resonance. In an appendix we indicate the steps which lead to the full mathematical model of quantum probability.
An outline of quantum probability
1 INDEX Introduction (1a.) Foundations of quantum theory (1b.) Quantum probability and the paradoxes of quantum theory (1c.) Von Neumann' s measurement theory (1d.) Contemporary measurement theory (1e.) Open systems and quantum noise (1f.) Stochastic calculus (1g.) Laws of large numbers and central limit theorems (1h.) Conditioning PART I: ALGEBRAIC PROBABILITY THEORY (2.) Algebraic probability spaces (3.) Algebraic random variables (4.) Stochastic Processes (5.) The local algebras of a stochastic process (6.) Independence (7.) Example: quantum spin systems (8.) A combinatorial lemma (9.) The Boson law of large numbers for independent random variables (10.) The central limit theorem for product maps (11.) Boson and Fermion Gaussian maps (12.) The quantum commutation relations as GNS representations (13.) The quantum commutation relations (14.) De Finetti' s theorem (15.) Conditioning: expected subalgebras (16.) Conditional amplitudes on B(H o ) (17.) Transition expectations and Markovian operators (18.) Markov chains, stationarity, ergodicity (19.) Conditional density amplitudes, potentials and invariant weights (20.) Multiplicative functionals and the discrete Feynman integral (21.) Quantum Markov chains and high temperature superconductivity models (22.) Kümmerer's Markov chains (23.) The algebraic states of Fannes, Nachtergaele and Slegers (24.) 1-dependence and the Ibragimov-Linnik conjecture (25.) 1-dependent quantum Markov chains 2 (26.) Commuting conditional density amplitudes (27.) Diagonalizable states (28.) A nonlinear chain of harmonic oscillators (29.) Generalized random walks (30.) The diffusion limit of the coherent chain (31.) Cecchini' s Markov chains PART II : STOCHASTIC CALCULUS (32.) Simple stochastic integrals (33.) Semimartingales and integrators (34.) Forward derivatives (35.) The o(dt)-notation (36.) Stochastic differential equations (37.) Meyer brackets and Ito tables (38.) The weak Itô formula (39.) The unitarity conditions (40.) The Boson Lévy theorem PART III : CONDITIONING (41.) The standard space of a von Neumann algebra (42.) The ϕ-conditional expectation
Classical Probability and Quantum Outcomes
Axioms, 2014
There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this conflict based on deriving commutativity from the weakest possible assumptions, and showing that stronger assumptions in some of the existing no-go proofs are unnecessary. An example of an unnecessary assumption in such proofs is an entangled system involving nonlocal observables. Another example involves the Kochen-Specker hidden variable model, features of which are also not needed to derive commutativity. A diagram is provided by which user-selected projectors can be easily assembled into many new, graphical no-go proofs.
Strongly coupled quantum and classical systems and Zeno's effect
Physics Letters A, 1993
A model interaction between a two-state quantum system and a classical switching device is analysed and shown to lead to the quantum Zeno effect for large values of the coupling costant κ. A minimal piecewise deterministic random process compatible with the Liouville equation is described, and it is shown that κ −1 can be interpreted as the jump frequency of the classical device.