nt - p h / 06 01 15 8 v 3 7 A ug 2 00 6 Quantum Probability Theory
نویسنده
چکیده
The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient. These illustrate the physical significance of the previously mentioned differences. ∗Work supported by the Hungarian Scientific Research Fund (OTKA); contract number: T 043642.
منابع مشابه
ua nt - p h / 06 01 15 8 v 3 7 A ug 2 00 6 Quantum Probability Theory
The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray,...
متن کاملua nt - p h / 01 12 06 1 v 3 6 A ug 2 00 2 Bogoliubov transformations and exact isolated solutions for simple non - adiabatic Hamiltonians
We present a new method for finding isolated exact solutions of a class of non-adiabatic Hamil-tonians of relevance to quantum optics and allied areas. Central to our approach is the use of Bogoliubov transformations of the bosonic fields in the models. We demonstrate the simplicity and efficiency of this method by applying it to the Rabi Hamiltonian.
متن کاملar X iv : q ua nt - p h / 05 08 06 7 v 1 8 A ug 2 00 5 NEW SCHEME OF QUANTUM TELEPORTATION
A new scheme for quantum teleportation is presented, in which the complete teleportation can be occurred even when an entangled state between Alice and Bob is not maximal.
متن کاملar X iv : q ua nt - p h / 06 11 18 7 v 1 1 7 N ov 2 00 6 Philosophical Aspects of Quantum Information Theory
2 First steps with quantum information 3 2.1 Bits and qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The no-cloning theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Quantum cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.1 Key Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Entanglement-as...
متن کامل