We generalize ordinary probability theory to those von Neumann algebras A, for which Dye's generalized version of the Radon-Nikodym theorem holds. This includes the classical case in which A is an Abelian von Neumann algebra generated by an observable or complete set of commuting observables. Via Gleason's theorem, this also includes the case of ordinary quantum mechanics, in which A=B(H) is the von Neumann algebra of all bounded operators on a separable Hilbert space H. Particular consideration is given to the concepts of conditioning, sufficient statistics, coarse-graining, and filtering.
We generalize ordinary probability theory to those von Neumann algebras A, for which Dye's generalized version of the Radon-Nikodym theorem holds. This includes the classical case in which A is an Abelian von Neumann algebra generated by an observable or complete set of commuting observables. Via Gleason's theorem, this also includes the case of ordinary quantum mechanics, in which A=B(H) is the von Neumann algebra of all bounded operators on a separable Hilbert space H. Particular consideration is given to the concepts of conditioning, sufficient statistics, coarse-graining, and filtering.
Abstract Philosophical accounts of quantum theory commonly suppose that the observables of a quantum system form a Type-I factor von Neumann algebra. Such algebras always have atoms, which are minimal projection operators in the case of quantum mechanics. However, relativistic quantum field theory and the thermodynamic limit of quantum statistical mechanics make extensive use of von Neumann algebras of more general types. This chapter addresses the question whether interpretations of quantum probability devised in the usual manner continue to apply in the more general setting. Features of non-type I factor von Neumann algebras are cataloged. It is shown that these novel features do not cause the familiar formalism of quantum probability to falter, since Gleason's Theorem and the Lüders Rule can be generalized. However, the features render the problem of the interpretation of quantum probability more intricate.
Introduction Quantum probability is a generalization of both classical probability theory and quantum mechanics that allows to describe the probabilistic aspects of quantum mechanics. This generalization is formulated in two steps. First, the theory is reformulated in terms of algebras of functions on probability spaces. Therefore, the notion of a probability space (Ω, ℱ , P ) is replaced by the pair ( L ∞ (Ω), E (.) = ∫ Ω . d P ) consisting of the commutative von Neumann algebra of bounded random variables and the expectation functional. Then, the commutativity condition is dropped. In this way we arrive at the notion of a (von Neumann) algebraic probability space ( N ,Φ) consisting of a von Neumann algebra N and a normal (faithful tracial) state Φ. As we have seen this includes classical probability spaces in the form ( L ∞ (Ω), E ), it also includes quantum mechanical systems modeled by a Hilbert space H and a pure state ψ ∊ H (or a mixed state ρ ∊ S ( H )), if we take N = B ( H ) and Ф the state defined by Ф( X ) = 〈 ψ, X ψ 〉 (or Ф ( X ) = tr(ρ X )) for X ∊ B ( H ). Note that in this course we shall relax the conditions on N and Φ and work with involutive algebras and positive normalized functionals, that is, *-algebraic probability spaces. A striking feature of quantum probability (also called noncommutative probability) is the existence of several notions of independence. This is the starting point of this course, which intends to give an introduction to the theory of quantum stochastic processes with independent increments. However, before we come to these processes, we will give a general introduction to quantum probability. In Section 1.2, we recall the basic definitions of quantum probability and discuss some fundamental differences between classical probability and quantum probability. In Section 1.3 we address the question ‘Why do we need Quantum Probability?’ We discuss the EPR experiment and the Kochen–Specker Theorem, which show that we cannot model quantum physics with classical probability spaces because the values of observable quantities do not exist unless we specify which measurement we will carry out and which quantities we will determine. In this sense quantum physics requires a more radical description of chance.
Introduction.The theory of information, created by Shannon [23], is developed by Feinstein, Kullback, MacMillan, Wiener and other American statisticians (e. g., cf.[10]), and also advanced into the ergodic theory by Gelfand, Khinchin, Kolmogorov, Yaglom and other Russian probabilists (e. g., cf.[8]).Through recent years, the theory is regarded as a new chapter in the theory of probability.Recently, Segal [22] gave a mathematical formulation of the entropy of state of a von Neumann algebra, which contains both the cases for the theory of information and the theory of quantum statistics.Segal's theorem was reformulated in operator algebraic form by Nakamura and Umegaki [16] and independently by Davis [3].Since the summer in 1954, Nakamura and Umegaki have investigated the concept of the conditional expectation in von Neumann algebra as a noncommutative extension of probability theory (cf.for example [13~18] and [25 ~28]), and in the most recent paper [18] it was applied to the theory of measurements of quantum statistics which is regarded as a non-commutative case of the theory of entropy and information.Furthermore, it may be very intersesting to develope the theory of information under functional-analysistic and operator-theoretic methods.From these points of views, we shall discuss the measure of information of integrable operators or of normal states of a von Neumann algebra.Davis [3] has independently studied on the almost same theme with Nakamura-Umegaki [16] and [18], in which he developed the theory of entropy and he simplified the proof of the theorem relative to the operatorentropy.Now, we shall give the basic notations and describe the fundamental concepts in a von Neumann algebra which will be used throughout the present paper.Let A be a von Neumann algebra, that is, A is a weakly closed self-adjoint algebra of bounded operators acting over a complex Hubert space H, which contains the identity operator 7. A linear functional p of A is said to be positive if p(aa*) ^ 0 for every a ε A. Such p is said to be state if p(I) = 1, to be normal in the terminology of Dixmier [4] if p(a a )tp(a) for a a ^a, and to be trace if p(ab) = p(ba) for every pair α, b ε A. The normality of state is equivalent to the complete additivity: Σ p(p a ) = p(Σ Pa) for any disjoint family of projections {p a }aA (cf.Dixmier [4]).
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.
The paper can be regarded as a short and informal introduction to noncommutative calculi of probability. The standard theory of probability is reformulated in the algebraic language. In this form it is readily generalized to that its version which is virtually present in quantum mechanics, and then generalized to the so-called free theory of probability. Noncommutative theory of probability is a pair (M, φ) where M is a von Neumann algebra, and φ a normal state on M which plays the role of a noncommutative probability measure. In the standard (commutative) theory of probability, there is, in principle, one mathematically interesting probability measure, namely the Lebesgue measure, whereas in the noncommutative theories there are many nonequivalent probability measures. Philosophical implications of this fact are briefly discussed.
In "Self-adjoint Operators as Functions I: Lattices, Galois Connections, and the Spectral Order" [arXiv:1208.4724], it was shown that self-adjoint operators affiliated with a von Neumann algebra N can equivalently be described as certain real-valued functions on the projection lattice P(N) of the algebra, which we call q-observable functions. Here, we show that q-observable functions can be interpreted as generalised quantile functions for quantum observables interpreted as random variables. More generally, when L is a complete meet-semilattice, we show that L-valued cumulative distribution functions and the corresponding L-quantile functions form a Galois connection. An ordinary CDF can be written as an L-CDF composed with a state. For classical probability, one picks L=B(Ω), the complete Boolean algebra of measurable subsets modulo null sets of a measurable space Ω. For quantum probability, one uses L=P(N), the projection lattice of a nonabelian von Neumann algebra N. Moreover, using some constructions from the topos approach to quantum theory, we show that there is a joint sample space for all quantum observables, despite no-go results such as the Kochen-Specker theorem. Specifically, the spectral presheaf Σ of a von Neumann algebra N, which is not a mere set, but a presheaf (i.e., a 'varying set'), plays the role of the sample space. The relevant meet-semilattice L in this case is the complete bi-Heyting algebra of clopen subobjects of Σ. We show that using the spectral presheaf Σ and associated structures, quantum probability can be formulated in a way that is structurally very similar to classical probability.