Three lectures will be presented on some matrix problems and techniques
in quantum information science. No quantum mechanics background
is required. Audience should have the basic linear algebra knowledge.
One may see the recent reprints and preprints of Li and his collaborators
on arXiv to get some ideas about the topics he plans to present.
Lecture 1 -- Optimization Problems in Quantum Information Science
In the mathematical framework, quantum states are represented
as density matrices, i.e., positive semidefinite matrices with
trace one, and quantum operations are represented as trace preserving
completely positive linear maps. In quantum information science,
one often has to estimate various measures between different
quantum states and quantum processes. One also needs to design
quantum operations with special properties. These give rise to
many optimization problems involving matrices and linear transformations.
In this lecture, selected results, techniques, and open problems
in this line of study will be described.
Lecture 2 -- Numerical Ranges and Quantum Information Science
In quantum mechanics, measurement operators or observable are
represented as Hermitian matrices, and measurement are done by
taking the inner product of the measurement operators and the
states (represented as density matrices). The collection of such
measurement values on states can be viewed as elements in the
joint numerical range of the measurement operators. Also, in
the study of quantum operations with special properties, and
the quantum error correction codes of quantum channels, one can
formulate the problems in terms of the higher rank numerical
ranges of the Choi-Kraus operators of the quantum operations/channels.
In this lecture, problems and results involving different kind
of numerical ranges will be described.
Lecture 3 -- Preserver Problems and Quantum Information Science
Preserver problems concern the characterization of maps on matrices
or operators with special properties. In connection to quantum
information science, researcher are interested in maps that leave
invariant some certain measures, relations, or subsets of quantum
states or quantum systems. In this lecture, selected problems
and results in such research will be described