J1-8132 Positive maps and real algebraic geometry

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Research project is (co) funded by the Slovenian Research Agency.

UL Member: Faculty of Mathematics and Physics
Code: J1-8132
Project: Positive maps and real algebraic geometry
Period: 1.5.2017 - 30.4.2020
Range per year: 1,03 FTE, category A
Head: Igor Klep
Research activity: Natural sciences and mathematics
Research Organisations: link on SICRIS
Researchers: link on SICRIS
Citations for bibliographic records: link on SICRIS

Project description:

A linear map phi between matrix algebras is positive if it maps positive semidefinite matrices to positive semidefinite matrices, and is completely positive (cp) if each of its ampliations I_kotimes phi is positive. (Completely) positive maps are ubiquitous in matrix theory, operator algebras, mathematical physics and quantum information theory. In the proposed project we shall investigate the gap between the sets of positive and cp maps. We conjecture that there are many more positive than completely positive maps. The proposal intends to combine in a novel way algebraic, geometric and analytic tools to settle and quantify the conjecture. In the project we also intend to find an algorithm for constructing positive maps that are not completely positive.

Work packages:

Many of the problems we address in this proposal are independent of each other, so the order in which we tackle them is flexible. We will work on the three strands of the proposal concurrently.

It is impossible to give a detailed plan for research that will be undertaken over three years in an area with lots of activity; we try to present our expected timetable below. We will publish our results in prestigious international journals and disseminate them at international conferences. In addition to the timeline presented below, we shall spend considerable effort, presumably during the summers (of years 2 and 3) on implementations of our algorithms and applications.