Block course on:
Quantum Information Theory in Infinite Dimensions: An Operator-Algebra Approach.
PD Prof. Michael Keyl (FU Berlin)
Most of quantum information theory is developed in the framework of finite
dimensional Hilbert spaces, and therefore not directly applicable to systems
like free and interacting non-relativistic particles, spin-systems in the
thermodynamic limit or relativistic field models, where an infinite
dimensional description is required. In some cases a more or less direct
generalization is possible (e.g. by replacing finite sums with absolutely
converging sequences) but this approach is very limited and misses many of
the more interesting aspects of infinite dimensional systems. In other words
mathematically and conceptually new tools are needed. In this context the
theory of operator algebras provides a very powerful framework, which is
particularly useful for the study of infinite degrees of freedom systems. The
purpose of this lecture series is to introduce into this theory and its
applications in qantum physics. Apart from the corresponding mathematical
foundations we will show how elementary concepts of quantum theory can be
reformulated and how the differences between finite dimensions, infinite
dimensions but finite degrees of freedom, and infinite degrees of freedom can
be related to operator algebras and their representations. Furthermore we
will study infinite spin systems, their entanglement properties and their
connection to advanced operator algebraic topics, like type and cassification
of von Neumann algebras.