Table of Contents / Contents

• Announcements
• Teaching Information
• Teaching Outline
• Textbooks and Other References
• To read before class
• Teaching Requirements and Exams
• Related Links

Announcements

• Nov. 16th: Course begins. Welcome reception at 9:00.
  
  
• Schedule: morning 9:00-12:15, afternoon 1:30-4:45 (30 hours over five days).
  
  
• Nov. 20th: Course ends.
  

Course Information

• Programme: ATHENS, a European exchange network of some fifteen universities: Partner institutions exchange their students for one intensive course week.
  
  
• Audience: Students of the ATHENS network and of Télécom Paris, from all fields; registration through the ATHENS programme of your home institution.
  
  
• Code: MOB_0AT47_TP
  
  
• Venue: Télécom Paris, 19 place Marguerite Perey, 91120 Palaiseau (Institut Polytechnique de Paris campus)
  
  
• Language: English
  
  
• Teaching Team: Prof. Olivier Rioul and Victor Rabiet (Invited Professor)
  
  
• Website: https://perso.telecom-paristech.fr/rioul/cours/quantum-athens.html
  
  
• Flyer: https://perso.telecom-paristech.fr/rioul/cours/quantum/flyer-athens.pdf

Course Outline

• Objective and Summary: This course offers a step-by-step introduction to the mathematical foundations of quantum information, starting from simple postulates within finite-dimensional systems.
  
  
• Who should attend?  The course is intended for a broad audience of enthusiastic students from all fields who wish to develop a solid theoretical understanding of the principles underlying quantum computing and quantum information.
  
  
• Prerequisites: As a prerequisite, participants are recommended to be familiar with linear algebra, complex numbers, and elementary probability theory and information theory. It should be noted that, although the course does not presuppose knowledge of Hilbert spaces, it does presuppose a fairly comprehensive knowledge of finite-dimensional complex scalar product spaces (also known as Hermitian spaces, the natural generalization of Euclidean spaces over the complex numbers).
  
  This course does not presuppose any knowledge of quantum physics but focuses on the mathematics of quantum information.
  
  
• Syllabus: This course offers a step-by-step introduction to the mathematical foundations of quantum information, starting from simple postulates within finite-dimensional systems.  A central emphasis is placed on quantum measurement and its inherently probabilistic nature, from which the linear evolution of quantum states and the derivation of the Schrödinger differential equation for isolated systems can be mathematically derived. 
  The course then addresses more realistic scenarios involving noise or interactions with an external environment.  In such settings, quantum states are described by density matrices, while their transformations are modelled by quantum channels, together with a general framework for quantum measurements. 
  If time permits, the course will also explore mathematical tools used to define and compare distances, divergences or similarities between quantum states, highlighting their connections with quantum channels and measurement processes.
  
  
• Main themes, in the order they are covered during the week:
  

1. States and measurement. The postulates of quantum mechanics in finite dimension; quantum measurement and its probabilistic nature; Hermitian (Hilbert) spaces, Dirac notations, spectral decompositions.
   
2. Observables and evolution. Observables and projective measurements; the uncertainty principle; quantum evolution via Wigner’s and Stone’s theorems, leading to the Schrödinger equation.
   
3. Multiple systems. Tensor products of Hermitian spaces and of operators; partial measurements and POVMs; cloning and discrimination of quantum states.
4. Mixed states. Partial and global trace, Schmidt decomposition and SVD; density operators, and how measurement and evolution extend to them.
5. Quantum channels and beyond. Quantum channels, classical versus quantum information; trace distance, fidelity and (if time permits) entropies.
   
   

Course Materials

• Handout: TBA  
• Bibliography:

1. John Watrous, Understanding Quantum Information and Computation: A Course on the Theory of Quantum Computing
   
   
2. Michael Nielsen and Isaac Chuang, Quantum Computation and Quantum Information
   
   
3. Olivier Rioul, Towards a Minimal Axiomatization of Quantum Information, GRETSI 2025  (contains proofs of Wigner's and Stone's theorems)
   

Study before class!

• Participants are recommended to be familiar with linear algebra, complex numbers, and elementary probability theory; they may want to review or expand their knowledge in the following mathematical areas (treated in class):
  
  
  
  • Hermitian spaces (finite dimensional Hilbert spaces)
    
    
  • Spectral decompositions of normal operators
    
    
  • Tensor product of spaces and of operators
    
    
  • Partial trace (vs. global trace) of operators
    
    
  • Singular value decomposition
    
    
  • Operator spaces (Hilbert-Schmidt inner product and Schatten norms)
    
    

Assessment

• Assessment Methods:  Exercises given in class + Attendance
  
  

Useful Links

• Quantum courses by IBM