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Representation Theory for Quantum Information Science — QMATH Masterclass 2025
These are the lecture notes (PDF) for my tutorial on representation theory with applications to quantum information science, delivered at the QMATH Masterclass on Representation Theory in Quantum Information Science in Copenhagen, 15 August 2025.
Structure of my lectures at the Masterclass
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Crash course on the basics of representation theory
Lecture notes (PDF)
Topics:- Groups, cosets, and actions
- Representations of finite groups and Maschke’s theorem
- Schur’s Lemma and commutants
- Character theory and orthogonality relations
- Fourier analysis on groups
- Schur–Weyl duality and applications in quantum information
- Quick overview of compact groups and tensor products
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Lectures on Haar measure tools for quantum information
Tutorial: Introduction to Haar Measure Tools in Quantum Information -
Lecture on the Clifford commutant
Based on our recent work A complete theory of the Clifford commutant
Slides (PDF)
Quantum Information, 2022-23 (Tutorials, FU Berlin)
Students can send me the week’s exercises (photos of the sheets or notes written on tablet) via e-mail . For my tutorial, the usual deadline for submitting the exercises is Sunday at 11:59 pm.
Email me if you would like to be added to the mailing list that I will use for quick communications related to my tutorials. This webpage can be useful for the students especially to have the tablet notes that I write and use for my tutorials.
Exercises
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Problem Sheet 1: “Density matrices and Bell experiments”. (PDF, Solutions)
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Problem Sheet 2: “POVMs and encoding classical information”. (PDF, Solutions)
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Problem Sheet 3: “Quantum Teleportation and p-norms”. (PDF, Solutions)
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Problem Sheet 4: “Graphical calculus and Quantum Channels”. (PDF, Solutions)
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Problem Sheet 5: “More Quantum Channels and Entropy”. (PDF, Solutions)
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Problem Sheet 6: “Operator Properties and LOCC”. (PDF, Solutions)
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Problem Sheet 7: “Capacities and Majorization”. (PDF, Solutions)
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Problem Sheet 8: “Entanglement Witnesses and Cryptography”. (PDF, Solutions)
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Problem Sheet 9: “Quantum Fourier Transform and Stabilizers”. (PDF, Solutions)
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Problem Sheet 10: “More stabilizers and quantum gates”. (PDF, Solutions)
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Problem Sheet 11: “Measurement based quantum computing”. (PDF, Solutions)
Here are some additional notes used in my tutorial:
- “PVM, POVM and Naimark dilatation theorem”. (PDF)
- “Quantum Channels Theorems”. (PDF)
- “Simple proofs of Schmidt and Kraus decomposition” (PDF).
- “Classical Entropies”. (PDF)
- “Quantum Entropies”. (PDF)
- “Basics of Quantum Computing”. (PDF)
- “Stabilizer formalism”. (PDF)
Theory
The main page of the course is here, where you can find the Lecture notes of the course written by Jens Eisert.
Other good Quantum Information notes are for example those written by Michael M. Wolf PDF, those written by John Preskill PDF and those written by Joseph M. Renes PDF.
The book written by Nielsen & Chuang can be found here.
A good reference for Information Theory is the book of Cover & Thomas PDF, while a good reference for Quantum Shannon Theory is Mark Wilde’s book PDF.