Prerequisites:
Basic linear algebra and basic (functional) analysis. Some knowledge
of quantum mechanics would be helpful.
Recommended Literature:
There will be no textbook. The following books
contain part of what I plan to cover in the course:
1) Robert Alicki and Mark Fannes, Quantum Dynamical Systems,
Oxford University Press, 2001.
2) R.F. Werner, Quantum Information Theory - An Invitation,
quant-ph/0101061.
3) Michael Nielsen, Isaac Chuang, Quantum Computation and Quantum
Information, Cambridge University Press, 2000.
Syllabus:
I will discuss in this course some of the mathematical concepts
and ideas that appear in quantum information theory and
quantum dynamical systems.
After a short introduction to quantum mechanics (observables, states,
measurements, density matrices and all that), I plan to discuss the
notion of entanglement of quantum systems. Entanglement is a feature
of quantum mechanics, which does not exist in classical physics. It
expresses a correlation of subsystems of a quantum physical system which
appears naturally as soon as the commutative algebras of functions in
classical physics are replaced by non-commutative algebras of operators
(matrices) in quantum physics. Entanglement is believed to be related to
what speeds up a quantum computer and is currently the subject of intense
study in quantum information science. If a correlated quantum system
contains ``enough'' entanglement, then it can be used to transmit quantum
information on a classical channel (quantum teleportation). Mathematically,
noncommutative structures described by operator theory, the
theory of operator algebras and the theory of completely
positive maps are at the heart of these phenomena.
Quantum dynamical systems are given by structure preserving maps of
noncommutative topological spaces or noncommutative measure spaces. If
time allows I will discuss such systems and present notions of entropy
(due to von Neumann, Connes and others) that are used to study mixing
properties of such maps. These notions of entropy turn out to be
very closely related to various measures which are currently proposed as a
way to quantify entanglement of quantum physical systems.
Grading: There will be no exams. The course grade will be based on
attendance. I will assign (optional) problems during the lectures.
Remark: I have given a course on quantum information theory in
Spring 2001.
Some overlap with my previous course is unavoidable (eg. the introductory
part to quantum mechanics might be similar), but the emphasis of my previous
course was on computing, quantum computing, quantum algorithms, error
correction etc.. The main topic of this course will not be
quantum computing, although the principles discussed in the course will
be motivated by quantum computing and will pertain to quantum computers
as a particular case of quantum physical
systems. I will emphasize in this course more the mathematical
structures underlying quantum information theory, entanglement and quantum
dynamical systems.
Remark: Atac Imamoglu is teaching a course on quantum information
processing in the spring quarter as well. To find out what he will
emphasize, please contact him at atac@ece.ucsb.edu.