A concept of conditional expectations in quantum theory is established with interrelations to previously introduced concepts of the Cycon–Hellwig conditional expectations and a posteriori states, which are analogous to the existing interrelations in the classical probability theory among conditional expectations related to random variables, those related to σ subalgebras and conditional probability distributions. These three concepts are shown to have satisfactory statistical interpretation in the quantum measuring processes. For the above purpose, we introduce an integration with respect to functions with values in the states of operator algebras and positive operator valued measures such that the resulting indefinite integrals are completely positive map valued measures. Eventually, it is proved that in the von Neumann algebraic formulation, the Cycon–Hellwig conditional expectations always exist as completely positive map valued measures.
A concept of conditional expectations in quantum theory is established with interrelations to previously introduced concepts of the Cycon–Hellwig conditional expectations and a posteriori states, which are analogous to the existing interrelations in the classical probability theory among conditional expectations related to random variables, those related to σ subalgebras and conditional probability distributions. These three concepts are shown to have satisfactory statistical interpretation in the quantum measuring processes. For the above purpose, we introduce an integration with respect to functions with values in the states of operator algebras and positive operator valued measures such that the resulting indefinite integrals are completely positive map valued measures. Eventually, it is proved that in the von Neumann algebraic formulation, the Cycon–Hellwig conditional expectations always exist as completely positive map valued measures.
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]).
In order to develop a statistical theory of quantum measurements including continuous observables, a concept of a posteriori states is introduced, which generalizes the notion of regular conditional probability distributions in classical probability theory. Its statistical interpretation in measuring processes is discussed and its existence is proved. As an application, we also give a complete proof of the Davies-Lewis conjecture that there are no (weakly) repeatable instruments for non-discrete observables in the standard formulation of quantum mechanics, using the notion of a posteriori states.
An extension of the conditional expectations (those under a given subalgebra of events and not the simple ones under a single event) from the classical to the quantum case is presented. In the classical case, the conditional expectations always exist; in the quantum case, however, they exist only if a certain weak compatibility criterion is satisfied. This compatibility criterion was introduced among others in a recent paper by the author. Then, state-independent conditional expectations and quantum Markov processes are studied. A classical Markov process is a probability measure, together with a system of random variables, satisfying the Markov property and can equivalently be described by a system of Markovian kernels (often forming a semigroup). This equivalence is partly extended to quantum probabilities. It is shown that a dynamical (semi)group can be derived from a given system of quantum observables satisfying the Markov property, and the group generators are studied. The results are presented in the framework of Jordan operator algebras, and a very general type of observables (including the usual real-valued observables or self-adjoint operators) is considered.
Expectations have been considered as dual objects of instruments in several papers on generalized probability theory and quantum theory. Here, we relate a generalized conditional (GC) expectation to a given instrument and a given state by two requirements, which are analogous to the axioms by which the classical conditional expectation is related to a given sub-σ-algebra and a given probability measure. In examples we illustrate the close similarity of GC expectations with classical conditional expectations. Eventually, we introduce a rich class of quantum stochastic processes, which are Markovian.
Abstract Given the state of a quantum system, we define the conditional expectation of an observable with respect to a second observable, or commuting family of observables, as the real-valued function of the latter which best approximates the observable, in the sense of minimizing the quantum mechanical expectation of the difference squared. This corresponds to the notion of best predictor of a random variable by another random variable in non-parametric statistical regression. A basic example is the conditional expectation of the quantum mechanical momentum with respect to position, which can be identified with the Bohm momentum. We re-examine the de Broglie–Bohm theory from this perspective. As a second example we consider the simplest quantum system, that of a spin- 1 / 2 paticle in a magnetic field. We study the conditional expectations of the Pauli spin-operators S 1 , S 2 , S 3 with respect to S 3 , and determine their dynamics if the pure-state wave-function solves the relevant Schrödinger equation. Considering S 3 as a beable we can, following Bell, introduce a generalized guidance law in the form of a Markov jump process on the spectrum of S 3 which is compatible with the time evolution of the Born probabilities of the two eigenstates of S 3 . The conditional expectations of the remaining spin operators are then shown to follow a piece-wise deterministic Markov process, defined by an explicit SDE. We finally examine the relationship between the conditional expectations of this paper, and the von Neumann-algebraic conditional expectation onto the sub-algebra generated by the commuting family of conditioning operators when the joint spectrum of the latter is discrete, and derive a general expression for the conditional expectation dynamics in that context.
Given a normalized state-vector ψ, we define the conditional expectation $\mathbb{E }_{ψ} (A | B ) $ of a Hermitian operator $A $ with respect to a strongly commuting family of self-adjoint operators $B $ as the best approximation, in the operator mean square norm associated to ψ, of $A $ by a real-valued function of $B . $ A fundamental example is the conditional expectation of the momentum operator $P $ given the position operator $X $, which is found to be the Bohm momentum. After developing the Bohm theory from this point of view we treat conditional expectations with respect to general $B $, which we apply to non-relativistic spin 1/2-particles. We derive the dynamics of the conditional expectations of momentum and spin with respect to position and the third spin component. These dynamics can be interpreted in terms of classical continuum mechanics as a two-component fluid whose components carry intrinsic angular momentum. Interpreting the joint spectrum of the conditioning operators as a space of beables, we can introduce a classical-stochastic particle dynamics on this space which is compatible with the time-evolution of the Born probability, by combining the de Broglie-Bohm guidance condition with a Markov jump process, following an idea of J. Bell. This results in a new Bohm-type model for particles with spin. A basic problem is that such auxiliary particle dynamics are far from unique. We finally examine the relation of our conditional expectations with the conditional expectations of the theory of $C^* $-algebras and, as an application, derive a general evolution equation for conditional expectations for operators acting on finite dimensional Hilbert spaces. Two appendices re-interpret the classical Bohm model as an integrable constrained Hamiltonian system, and provide the details of the two-fluid interpretation.