GAMP/QMATH Lecture: Containment between LMI domains and the existence of quantum channels
Title: Containment between LMI domains and the existence of quantum channels
Abstract:"In convex optimization, quantum information theory and real 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). However, even when we wish to check for containment of the n-dimensional cube inside some other LMI domain, this is an NP-hard problem.
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 discuss some geometric aspects of these relaxed containment problems. We will explain how to move the original problem to the relaxed problem, how to estimate the error in this passage, connections with the existence of quantum channels and how in some cases we can "rescale" a unital positive map to an entanglement breaking map.
*Based on joint work with Kenneth R. Davidson, Orr Moshe Shalit and Baruch Solel."