ICNCP
Independence and Convolutions in Noncommutative Probability
| Coordinatore | UNIVERSITE DE FRANCHE-COMTE
Organization address
address: CLAUDE GOUDIMEL 1 contact info |
| Nazionalità Coordinatore | France [FR] |
| Totale costo | 194˙046 € |
| EC contributo | 194˙046 € |
| Programma | FP7-PEOPLE
Specific programme "People" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013) |
| Code Call | FP7-PEOPLE-2012-IIF |
| Funding Scheme | MC-IIF |
| Anno di inizio | 2013 |
| Periodo (anno-mese-giorno) | 2013-04-01 - 2015-03-31 |
Partecipanti
| # | ||||
|---|---|---|---|---|
| 1 |
UNIVERSITE DE FRANCHE-COMTE
Organization address
address: CLAUDE GOUDIMEL 1 contact info |
FR (BESANCON) | coordinator | 194˙046.60 |
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Obiettivo del progetto (Objective)
'Noncommutative probability, also called quantum probability or algebraic probability theory, is an extension of classical probability theory where the algebra of random variables is replaced by a possibly noncommutative algebra. A surprising feature of noncommutative probability is the existence of many very different notions of independence. The most prominent among them is freeness or free probability, which was introduced by Voiculescu to study questions in operator algebra theory. In the last twenty-five years, free probability has turned into a very active and very competitive research area, in which analogues for many important probabilistic notions like limit theorems, infinite divisibility, and L'evy processes have been discovered. It also turned out to be closely related to random matrix theory, which has important applications in quantum physics and telecommunication.
The current project proposes to study the mathematical theory of independence in noncommutative probability, and the associated convolution products. We will concentrate on the following topics:
(1) Applications of monotone independence to free probability. Some applications have been found already, but recent work indicates that much more is possible.
(2) Analysis of infinitely divisible distributions in classical and free probability. Common complex analysis methods will be used for both classes, and we expect more insight into their mutual relations.
(3) Application and development of Lenczewski's matricial free independence. This concept introduces very new ideas, whose better understanding will certainly lead to new interesting results.
The methods we will use in this project come not only from noncommutative probability, but also from functional analysis, complex analysis, combinatorics, classical probability, random matrices, and graph theory.'
Introduzione (Teaser)
EU-funded research has made important progress in deriving mathematical descriptions of quantum probability, an extension of classical probability that has important applications in quantum physics and telecommunication.
Descrizione progetto (Article)
The main focus of probability theory is the algebra of random variables. In the conventional approach, one selects a sample space and assigns a probability (expectation) to a number of events in that space, building algebras of random variables. The random variables, the probability of a certain event in that sample space, are commutable, meaning that changing the order of operands does not change the result.
In quantum mechanics, the sample space is replaced by the space of states and the expectation is now the expected value on a given quantum state. Physical observables take the place of the random variables that generally do not commute. Quantum probability, also called noncommutative probability, incorporates the possibility of noncommutative operations, encompassing both the quantum and classical states. Developed in the 1980s, it has provided models of quantum observation processes that resolve many of the apparent inconsistencies of quantum mechanics.
Quantum probability contains many different notions of independence, the most prominent of which is free probability, a concept created around 1985. The discovery in 1991 that it is closely related to random matrix theory led to exciting new results, concepts and tools, and the identification of important applications. The EU-funded project 'Independence and convolutions in noncommutative probability' (ICNCP) investigated the mathematical theory of free probability and free independence, pushing the frontiers of both classical and free probability.
The short two-year research effort resulted in nine publications and 10 presentations. Outcomes will make an important contribution to the field and eventually to the description and development of practical devices.