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.
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.

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.

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.

Theory of entanglement.
Separable states; the LOCC paradigm; majorization and entanglement
transformation; measures of entanglement; the partial transpose and
bound entanglement.

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.
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:

M. Nielsen and I. Chuang.
Quantum Computation and Quantum Information.
Cambridge University Press, 2000.

M. Wilde.
Quantum Information Theory.
Second edition.
Cambridge University Press, 2017.

A. Kitaev, A. Shen, and M. Vyalyi.
Classical and Quantum Computation,
volume 47 of Graduate Studies in Mathematics.
American Mathematical Society, 2002.

R. Bhatia.
Matrix Analysis. Springer, 1997.

R. Horn and C. Johnson.
Matrix Analysis.
Cambridge University Press, 1985.
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.