Finite Quantum Instruments

TL;DR

This paper explores the structure and properties of finite quantum instruments and observables, including their combinations, compatibility, and measurement models, within finite-dimensional quantum systems.

Contribution

It introduces a comprehensive framework for finite quantum instruments, extending observable concepts and analyzing their interactions and measurement models.

Findings

Defined finite observables and instruments.

Analyzed combinations and relations of observables and instruments.

Presented various types of instruments and their measurement implications.

Abstract

This article considers quantum systems described by a finite-dimensional complex Hilbert space $H$. We first define the concept of a finite observable on $H$. We then discuss ways of combining observables in terms of convex combinations, post-processing and sequential products. We also define complementary and coexistent observables. We then introduce finite instruments and their related compatible observables. The previous combinations and relations for observables are extended to instruments and their properties are compared. We present four types of instruments; namely, identity, trivial, L\"uders and Kraus instruments. These types are used to illustrate different ways that instruments can act. We next consider joint probabilities for observables and instruments. The article concludes with a discussion of measurement models and the instruments they measure.