International workshop on "Quantum Physics and Geometry" |

PhD Course:

Quantum information theory and geometry

Quantum information theory and geometry

JM Landsberg (Texas A&M, USA)

Quantum information theory is at the intersection of physics, computer science and mathematics. While it is little understood, what we know already may have a dramatic impact on our lives. Efforts are now being made to build a quantum computer, which, if successful, would in particular end the security of public key cryptography among many other things. At the present, we have very little knowledge of what a quantum computer can and cannot compute efficiently. Much research remains to be done.

This course will discuss the physics of information, and the mathematics needed to study it. The first half will cover background material from physics and computer science. It will give an overview and foundation of the subject. The second half will be an in-depth study of geometry and representation theory useful for quantum information theory. The mathematics covered will prepare students to do research in this exciting and important area.

The primary target of the lectures are first year PhD students. However given its timely and interdisciplinary nature, it should also attract students and faculty in mathematics, physics and computer science.

Topics (approximately 5 hours of lecture each):

- Classical information theory (Shannon capacity, complexity).
- Quantum information theory and its connections to probability.
- Quantum complexity and algorithms.
- Representation theory for quantum information
- Algebraic geometry for quantum information.

Schedule of the course:

- Week 1 [June 5-9]: Background - classical information (Shannon
theory), background in quantum mechanics, probability and representation
theory,

basic definitions. - Week 2 [June 27-30]: Quantum computing.
- Week 3 [July 10-14]: Quantum information
theory.

Material for the course will be drawn from several sources, including:

- J. Preskill, Lecture notes on Quantum Information Theory

http://www.theory.caltech.edu/~preskill/ph229/#lecture

- Kitaev, A. Yu.; Shen, A. H.; Vyalyi, M. N.

Classical and quantum computation.

Translated from the 1999 Russian original by Lester J. Senechal. Graduate Studies in Mathematics, 47. American Mathematical Society, Providence, RI, 2002. xiv+257 pp. ISBN: 0-8218-2161-X

- Landsberg, J. M.; Tensors: geometry and applications. Graduate Studies in Mathematics, 128. American Mathematical Society, Providence, RI, 2012. xx+439 pp. ISBN: 978-0-8218-6907-9

First week:

- June 5th: (First Lecture not available), (Second Lecture)
- June 6th: (First Lecture), (Second Lecture)
- June 7th: (First Lecture), (Second Lecture)
- June 8th: (First Lecture), (Second Lecture)
- June 9th: (First Lecture not available), (Second Lecture)
- June 27th: (First Lecture), (Second Lecture)
- June 28th: (First Lecture), (Second Lecture)
- June 29th: (First Lecture), (Second Lecture)
- June 30th: (First Lecture), (Second Lecture)
- July 10th: (First Lecture), (Second Lecture)
- July 11th: (First Lecture), (Second Lecture)
- July 12th: (First Lecture not available), (Second Lecture)
- July 13th: (First Lecture), (Second Lecture)
- July 14th: (First Lecture), (Second Lecture)