Projects / Programmes
Positive maps and real algebraic geometry
| Code |
Science |
Field |
Subfield |
| 1.01.04 |
Natural sciences and mathematics |
Mathematics |
Algebra |
| Code |
Science |
Field |
| P001 |
Natural sciences and mathematics |
Mathematics |
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
real algebraic geometry; positive map; operator algebra; copositive matrix
Researchers (12)
Organisations (2)
Abstract
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.
Significance for science
The proposed project is in a currently very active field of mathematical research, so that the results achieved are important for the development of mathematical sciences. Studying positive and completely positive maps is an important part of linear algebra, functional analysis and operator algebra. Through the transition to polynomials we will connect these areas with (real and complex) algebraic geometry, which is a new approach to the study of positive maps. Completely positive maps are characterized by their Choi-Kraus representations, which enables efficient computations through semidefinite programming. On the other hand, very little is understood about positive maps, and verifying positivity of a map is NP-hard. This gives rise to very important questions, such as what is the gap between positive and completely positive maps and how well can one approximate positive maps with completely positive ones or such for which positivity can be effectively checked, for example by a semidefinite program. All these issues will be dealt with in our project. An important aspect of our project is also the construction of positive maps that are not completely positive. So far few such maps are known, although such transformations are fundamental in quantum information theory to distinguish separable and entangled states. Our results will allow the construction of many positive maps that are not completely positive, and this will in turn enable the development of new methods for the determination of the entanglement conditions in quantum information theory. Our research will also be associated with mathematical optimization, which is due to its applications one of the most active mathematical areas in recent years. Our results will contribute in particular to a better understanding of copositive optimization and semidefinite programming.
Significance for the country
Optimisation is ubiquitous in the sciences as well as in the economy. We estimate that the project's most direct impact on the economy will result from copositivity, as many well-known and in practice widely used algorithms for optimization of tough problems translate to copositive optimization. Additional impact is expected from our results in the fields of semidefinite programming and polynomial optimization. The project will develop new optimization methods and study their complexity and accuracy. Furthermore, our study is oriented to algorithmic methods of implementation, which are immediately useful in many fields of science, engineering, economics and elsewhere.
Most important scientific results
Interim report,
final report
Most important socioeconomically and culturally relevant results
Interim report,
final report
