IQIS Colloquium

The capacity of the classical channel was investigated by Claude Shannon in 1948. This capacity is additive under the tensor products of two channels. The capacity of a quantum channel (QC) was introduced by Alexander Holevo in 1998.

I will introduce a family of operator norms based on the Schmidt decomposition theorem and focus on their applications in quantum information theory. We will discuss two problems in particular: the existence problem for non-positive partial transpose bound entangled states and the problem of computing or estimating the minimum gate fidelity of a quantum channel. We will discuss some techniques for computing the norms and how they apply in the special cases of NPPT bound entanglement and minimum gate fidelity.

The physics literature in the last 10 years contains an immense number of papers related to multi-particle entanglement involving some very sophisticated mathematics. In this lecture we will survey some of this work that centers on geometric invariant theory.

The importance of information technology motivates fundamental questions about absolute limits to information itself. I show how laws of heat, disorder and gravitational collapse reveal the ultimate limits to information storage and behavior of complex systems according to a "holographic" correspondence between natural laws of our universe with the laws of a boundary for a higher-dimensional universe.

In recent years the study of tensors, i.e. multiarrays, became a topic of an extensive research in applied and pure mathematics. In this talk we discuss a number of problems in: quantum information theory, theoretical computer science, algebraic geometry and algebraic statistics.