---
title: Overview
description: Dive deeper into quantum information, including density matrices, channels, and general measurements.
source: https://eu-de.quantum.cloud.ibm.com/learning/en/courses/general-formulation-of-quantum-information
---

# Overview

!["Hero image"](https://eu-de.quantum.cloud.ibm.com/learning/images/courses/general-formulation-of-quantum-information/hero.svg)

Welcome to *General Formulation of Quantum Information,* the third course in the *Understanding Quantum Information and Computation* series comprising the following courses:

- [Basics of quantum information](/learning/courses/basics-of-quantum-information)
- [Fundamentals of quantum algorithms](/learning/courses/fundamentals-of-quantum-algorithms)
- General formulation of quantum information (this course)
- [Foundations of quantum error correction](/learning/courses/foundations-of-quantum-error-correction)

This course describes the general formulation of quantum information, where quantum states are represented by *density matrices,*
changes in states are described by *channels,* and a more general class of *measurements* can be considered than those discussed in
*Basics of quantum information.* The course also discusses mathematical ways of formalizing the *distance* or *similarity* between
quantum states, and how they relate to channels and measurements in operational ways.

This course is intended for students, professionals, and hobbyists in fields such as computer science, physics, engineering, and
mathematics who are eager to gain knowledge on the theoretical foundations of quantum information and computation.

## Exam

To earn your badge for General formulation of quantum information, take the exam at IBM® Training. This exam is intended to be taken after reading the lessons in this course. After you pass the exam, you will be notified by Credly that you have earned a badge.

[Exam](/learning/courses/general-formulation-of-quantum-information/exam)

## Awarded Badge

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## Contents

- [Overview](/learning/courses/general-formulation-of-quantum-information)
- Lessons
  - Density matrices
    - [Introduction](/learning/courses/general-formulation-of-quantum-information/density-matrices/introduction)
    - [Density matrix basics](/learning/courses/general-formulation-of-quantum-information/density-matrices/density-matrix-basics)
    - [Convex combinations of density matrices](/learning/courses/general-formulation-of-quantum-information/density-matrices/convex-combinations)
    - [Bloch sphere](/learning/courses/general-formulation-of-quantum-information/density-matrices/bloch-sphere)
    - [Multiple systems and reduced states](/learning/courses/general-formulation-of-quantum-information/density-matrices/multiple-systems)
  - Quantum channels
    - [Introduction](/learning/courses/general-formulation-of-quantum-information/quantum-channels/introduction)
    - [Quantum channel basics](/learning/courses/general-formulation-of-quantum-information/quantum-channels/quantum-channel-basics)
    - [Channel representations](/learning/courses/general-formulation-of-quantum-information/quantum-channels/representations-of-channels)
    - [Equivalence of the representations](/learning/courses/general-formulation-of-quantum-information/quantum-channels/representation-equivalence)
  - General measurements
    - [Introduction](/learning/courses/general-formulation-of-quantum-information/general-measurements/introduction)
    - [Mathematical formulations of measurements](/learning/courses/general-formulation-of-quantum-information/general-measurements/formulations-of-measurements)
    - [Naimark's theorem](/learning/courses/general-formulation-of-quantum-information/general-measurements/naimark-theorem)
    - [Quantum state discrimination and tomography](/learning/courses/general-formulation-of-quantum-information/general-measurements/discrimination-and-tomography)
  - Purifications and fidelity
    - [Introduction](/learning/courses/general-formulation-of-quantum-information/purifications-and-fidelity/introduction)
    - [Purifications](/learning/courses/general-formulation-of-quantum-information/purifications-and-fidelity/purifications)
    - [Fidelity](/learning/courses/general-formulation-of-quantum-information/purifications-and-fidelity/fidelity)
  - [Exam](/learning/courses/general-formulation-of-quantum-information/exam)