Description

This course presents a mathematical treatment of the theory of quantum information, with a focus on the development of concepts and methods that are fundamental to a broad range of studies in quantum algorithms and complexity, quantum cryptography, and quantum Shannon theory.

The course is intended for graduate students at the Masters or PhD level that have previously taken an introductory course (at the undergraduate or graduate level) in quantum computation.

Topics

The course will cover the following topics. The number of lectures on each topic varies with the material, and additional topics may be added as time permits.

  1. Fundamentals of quantum information. Fundamentals of quantum information. Quantum states, measurements, and channels, including representations and characterizations of measurements and channels; similarity and distance measures for states.
  2. Basic quantum Shannon theory. Von Neumann entropy and noiseless quantum coding; properties of von Neumann entropy and quantum relative entropy; strong subadditivity of von Neumann entropy and Holevo's theorem.
  3. Theory of entanglement. Separable states; the LOCC paradigm; majorization and entanglement transformation; measures of entanglement; the partial transpose and bound entanglement.
  4. Additional topics (as time permits). Semidefinite programming in quantum information theory; the completely bounded trace norm; unitarily invariant measures; the finite quantum de Finetti theorem; approximate cloning of pure states.

Reference material

The primary reference material is a draft of a book to be provided by the instructor. The following books represent optional additional sources of information:

Grading

Course grade is based 80% on homework assignments and 20% on a final project. Four problem sets will be assigned at regular intervals throughout the course.