Colloquium: Tuesday, March 27, 2 pm. Speaker: Adam Dor-On (Technion). Title: “Containment between LMI domains and the existence of quantum channels”.

In convex optimization, quantum information theory and real convex algebraic geometry, many practical and theoretical questions are related to containment problems between convex sets defined by a linear matrix inequality (LMI domains for short). One difficulty when we wish to check for containment of the n-dimensional cube inside some other LMI domain, is that this is computationally hard (NP-hard complexity).

These sort of problems become computationally tractable when we relax them to containment problems between *matrix* LMI domains. In fact, this relaxation of the problem enables the use of a semidefinite program to check for matrix LMI domain containment.

In this talk we will survey some of the geometric aspects of these relaxations. We will explain how to move the original problem to the relaxed problem, the connections with the existence of quantum channels, and how in some cases we can get an estimate (which is sometimes sharp) for the error of passing from the original LMI containment problem to the relaxed matrix LMI containment problem.

*Joint work with Kenneth R. Davidson, Orr Moshe Shalit and Baruch Solel.