Stan Gudder

Sequential quantum measurements

A quantum effect is an operator A on a complex Hilbert space H that satisfies 0⩽A⩽I. We denote the set of quantum effects by E(H). The set of self-adjoint projection operators on H corresponds to sharp effects and is denoted by P(H). We define the sequential product of A,B∈E(H) by A∘B=A1/2BA1/2. The main purpose of this article is to study some of the algebraic properties of the sequential product. Many of our results show that algebraic conditions on A∘B imply that A and B commute for the usual operator product. For example, if A∘B satisfies certain distributive or associative laws, then AB=BA. Moreover, if A∘B∈P(H), then AB=BA and A∘B=B or B∘A=B if and only if AB=BA=B. A natural definition of stochastic independence is introduced and briefly studied.

Sequential products on effect algebras

Abstract A sequential effect algebra (SEA) is an effect algebra on which a sequential product with natural properties is defined. The properties of sequential products on Hilbert space effect algebras are discussed. For a general SEA, relationships between sequential independence, coexistence and compatibility are given. It is shown that the sharp elements of a SEA form an orthomodular poset. The sequential center of a SEA is discussed and a characterization of when the sequential center is isomorphic to a fuzzy set system is presented. It is shown that the existence, of a sequential product is a strong restriction that eliminates many effect algebras from being SEAs. For example, there are no finite nonboolean SEAs, A measure of sharpness called the sharpness index is studied. The existence of horizontal sums of SEAs is characterized and examples of horizontal sums and tensor products are presented.

Mathematical Theory of Duality Quantum Computers

We present a mathematical theory for a new type of quantum computer called a duality quantum computer that has recently been proposed. We discuss the nonunitarity of certain circuits of a duality quantum computer and point out a paradoxical situation that occurs when mixed states are considered. It is shown that a duality quantum computer can measure itself without needing a separate measurement apparatus to determine its final state.

Sequentially independent effects

A quantum effect is a yes-no measurement that may be unsharp. An effect is represented by an operator E on a Hilbert space that satisfies 0 < E < I. We define effects E 1 , E 2 ,…,, E n to be sequentially independent if the result of any sequential measurement of E 1 , E 2 ,… E n does not depend on the order in which they are measured. We show that two effects are sequentially independent if and only if they are compatible. That is, their corresponding operators commute. We also show that three effects are sequentially independent if and only if all permutations of the product of their corresponding operators coincide. It is noted that this last condition does not imply that the three effects are mutually compatible.

INFIMA OF HILBERT SPACE EFFECTS

The quantum effects for a physical system are usually described by the set ℰ(H) of positive operators on a Hilbert spaceH that are bounded above by the unit operator. Under a natural order, ℰ(H) becomes a partially ordered set that is not a lattice unless dimH ⩽ 1. A characterization of the pairsA, B ɛ ℰ(H) such that the infimumA ∧ B exists is an open problem called the infimum problem. Methods of linear algebra are employed to solve the infimum problem when dimH < ∞. These methods do not appear to generalize to infinite dimensions so the general problem remains open. Progress toward solution of the general problem is discussed.

Effects as Functions on Projective Hilbert Space

The set of effect operators in a complex Hilbert space can be injectively embedded into the set of functions from the set of one-dimensional projections to the real interval [0, 1]. Properties of this injection are investigated.

Quantum measure and integration theory

This article begins with a review of quantum measure spaces. Quantum forms and indefinite inner-product spaces are then discussed. The main part of the paper introduces a quantum integral and derives some of its properties. The quantum integral’s form for simple functions is characterized and it is shown that the quantum integral generalizes the Lebesgue integral. A bounded, monotone convergence theorem for quantum integrals is obtained and it is shown that a Radon–Nikodym-type theorem does not hold for quantum measures. As an example, a quantum-Lebesgue integral on the real line is considered.

Quantum measure theory

We first present some basic properties of a quantum measure space. Compatibility of sets with respect to a quantum measure is studied and the center of a quantum measure space is characterized. We characterize quantum measures in terms of signed product measures. A generalization called a super-quantum measure space is introduced. Of a more speculative nature, we show that quantum measures may be useful for computing and predicting elementary particle masses.

Characterization of the sequential product on quantum effects

We present a characterization of the standard sequential product of quantum effects. The characterization is in term of algebraic, continuity and duality conditions that can be physically motivated.

An anhomomorphic logic for quantum mechanics

Although various schemes for anhomomorphic logics for quantum mechanics have been considered in the past we shall mainly concentrate on the quadratic or grade-2 scheme. In this scheme, the grade-2 truth functions are called coevents. We discuss properties of coevents, projections on the space of coevents and the master observable. We show that the set of projections forms an orthomodular poset. We introduce the concept of precluding coevents and show that this is stronger than the previously studied concept of preclusive coevents. Precluding coevents are defined naturally in terms of the master observable. A result that exhibits a duality between preclusive and precluding coevents is given. Some simple examples are presented.