Guide to Mathematical Concepts of Quantum Theory

TL;DR

This paper provides a comprehensive, mathematically detailed review of quantum theory's core concepts, including states, effects, measurements, channels, and entanglement, aimed at clarifying the modern language of quantum information and measurement.

Contribution

It offers a clear, step-by-step mathematical exposition of quantum theory concepts, integrating examples and exercises to enhance understanding of quantum experiments and information theory.

Findings

Detailed mathematical framework for quantum states and effects

Explanation of quantum measurement models and instruments

Overview of quantum entanglement properties

Abstract

Quantum Theory is one of the pillars of modern science developed over the last hundred years. In this review paper we introduce, step by step, the quantum theory understood as a mathematical model describing quantum experiments. We start with splitting the experiment into two parts: a preparation process and a measurement process leading to a registration of a particular outcome. These two ingredients of the experiment are represented by states and effects, respectively. Further, the whole picture of quantum measurement will be developed and concepts of observables, instruments and measurement models representing the three different descriptions on experiments will be introduced. In the second stage, we enrich the model of the experiment by introducing the concept of quantum channel describing the system changes between preparations and measurements. At the very end we review the elementary properties of quantum entanglement. The text contains many examples and exercise covering also many topics from quantum information theory and quantum measurement theory. The goal is to give a mathematically clear and self-containing explanation of the main concepts of the modern language of quantum theory.