Sequential Products of Quantum Measurements

TL;DR

This paper develops a comprehensive framework for sequential products involving quantum effects, operations, observables, and instruments, extending existing concepts and exploring their properties, conditioning, and coexistence in finite-dimensional Hilbert spaces.

Contribution

It introduces the novel concept of sequential products between effects and operations, and generalizes these to include observables and instruments, enriching the mathematical structure of quantum measurement theory.

Findings

Defined sequential products of effects with operations and vice versa.

Extended the framework to include observables and instruments.

Analyzed properties, conditioning, and coexistence of these quantum measurement constructs.

Abstract

Our basic structure is a finite-dimensional complex Hilbert space $H$. We point out that the set of effects on $H$ form a convex effect algebra. Although the set of operators on $H$ also form a convex effect algebra, they have a more detailed structure. W introduce sequential products of effect and operations. Although these have already been studied, we introduce the new concept of sequential products of effects with operations and operations with effects. We then consider various special types of operations. After developing properties of these concepts, the results are generalized to include observables and instruments. In particular, sequential products of observables with instruments and instruments with observables are developed. Finally, we consider conditioning and coexistence of observables and instruments.