Entropy of automorphisms
This edition contains one complete lesson: Operator convex functions and the continuity of entropy.
Learn operator convexity and concavity, Jensen’s operator inequality, Shannon entropy, eigenvalue inequalities and the sharp Fannes–Audenaert continuity bound. Includes full proofs, equality cases and three exercises with solutions.
Read the complete lesson · Download the reader and editable sources
The broader course is not complete in this edition. Only the lesson listed here is included.
Lesson contents
- Introduction
- Results used from other lessons
- 1. Operator convex functions
- 2. The functions \(t\log t\) and \(\log t\)
- 3. Jensen's operator inequality
- 4. Shannon entropy
- 5. The sharp continuity bound for Shannon entropy
- 6. Eigenvalue inequalities
- 7. The Fannes–Audenaert inequality
- 8. Exercises
- References
Authorship and preparation
Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).
Begin with the lesson’s five precisely stated prerequisite results: continuous functional calculus, the spectral theorem, positive maps on abelian algebras, Stinespring’s theorem and faithful representations. Each links to its full proof in the public programme.
English · original exposition under CC0 1.0 · human mathematical sources remain credited in the lesson.