Learn about quantum information theory and distance measures between states in this graduate-level lecture from the Park City Mathematics Institute's summer program on Quantum Computation. Explore fundamental concepts in quantum state discrimination and information transmission through quantum channels, building upon Shannon's classical information theory foundations. Delve into mathematical frameworks for studying quantum noise and entropy measures as part of a comprehensive series examining the computational power of near-term quantum devices. The lecture, delivered by Omar Fawzi from École Normale Supérieure de Lyon, includes accompanying lecture notes and problem sets to develop practical understanding of the material through guided exercises.