S. Gudder

Fixed points of quantum operations

Quantum operations frequently occur in quantum measurement theory, quantum probability, quantum computation, and quantum information theory. If an operator A is invariant under a quantum operation φ, we call A a φ-fixed point. Physically, the φ-fixed points are the operators that are not disturbed by the action of φ. Our main purpose is to answer the following question. If A is a φ-fixed point, is A compatible with the operation elements of φ? We shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes. Our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von Neumann algebras.

Noncommutative Probability on von Neumann Algebras

We generalize ordinary probability theory to those von Neumann algebras A, for which Dyes 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 Gleasons 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.

Observables and the Field in Quantum Mechanics

Corresponding to any irreducible proposition system L in general quantum mechanics there is a division ring D with an anti‐automorphism * and a vector space (V, D) over D with a definite sesquilinear form φ such that L is isomorphic to the set of φ closed subspaces of (V, D). The main task remaining in connecting the general quantum mechanics to the conventional quantum theory in a complex Hilbert space is to give physical arguments which force D to be the complex field. In this paper it is shown that if L admits a certain type of observable (together with other structure which seems to be physically justified), then D contains the real field as a subfield. Steps are then indicated that can be taken to move from the reals to the complexes or quaternions.

Generalized monotone convergence and Radon–Nikodym theorems

A measure and integration theory is presented in the quantum logic framework. A generalization of the monotone convergence theorem is proved. Counterexamples are used to show that the dominated convergence theorem, Fatou’s lemma, Egoroff’s theorem, and the additivity of the integral do not hold in this framework. Finally, a generalization of the Radon–Nikodym theorem is proved.

Measurements, Hilbert space and quantum logics

We consider single and multiple measurements on a quantum logic (P,S) as well as states and propositions conditioned by a measurement. We show that corresponding to any measurement A, there is a canonically associated Hilbert space HA. Algebraic and statistical properties of (P,S) that are preserved in HA are found. We then study the problem of embedding a quantum logic in Hilbert space.

Regular quantum Markov processes

In a previous work, quasidiscrete quantum Markov processes were considered. In order to describe certain continuum situations, the concept of a regular quantum Markov process is now developed. First, the general theory of such processes will be presented. Then, methods for constructing these processes will be considered. To accomplish this, the classical construction of measures on trajectory spaces to complex measures is generalized. A class of processes that have an associated family of transition amplitude operators is constructed. The paper concludes with various examples that illustrate the theory.