GWFP · Geometric study of Wasserstein spaces and free probability
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2019-10-01 → 2021-09-30
- EU contribution
- €186,167
- Participants
- 1
- Scheme
- MSCA-IF-EF-ST
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Results in brief
Geometric study of Wasserstein spaces and free probability
We studied geometric aspects of optimal transportation (OT) and quantum information (QI). Albeit the first version of the OT problem was formulated in the 18th century by Monge, the theory of OT became one of the central topics of analysis only ca. 20 years ago with intimate links to mathematical physics, theory of PDE's, and probability. The importance of OT theory was highlighted by two Fields-medals (Cedric Villani and Alessio Figalli) in the past decade. Our main objective concerning OT was the study of the metric structure of classical Wasserstein spaces with a strong emphasis on the classification of distance-preserving maps. Information theoretical dissimilarity measures (in short: divergences) play a central role in QI theory, and the geometric aspects of QI have been studied with a particular intensity in the last decades. Divergences play an essential role in quantum state tomography and quantum process tomography, quantum error correction, and quantum hypothesis testing. We studied the geometry of quantum states equipped with quantum Jensen-Shannon divergence (QJSD), which is the symmetric and bounded version of the von Neumann relative entropy. Our main objectives concerning QI were to prove the long-standing conjecture on the metric property of the QJSD, to give a center-of-mass interpretation of Kubo-Ando operator means, and to study quantum Hellinger distances. Impact for society: OT is a very natural problem with deep connections to other fields of science: economics, physics, etc. Our results concerning QI have the potential to be applied in quantum computing.
Data: CORDIS, © European Union
Project objective
The proposed research is divided into two main work packages. The first one is the study of spaces of measures equipped with the optimal transport distance (Wasserstein distance) with a special emphasis on the structure of isometries (surjective distance-preserving maps) and isometric embeddings (not necessarily surjective transformations that preserve the distance) of these spaces. The second work package is devoted to the investigation of measures from the viewpoint of free probability theory. This work package covers three subtopics: the qualitative behaviour of the free convolution, new random matrix ensembles arising from tensor networks, and the study of free Wasserstein spaces.
Original text from CORDIS.
Participants
- INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA · KlosterneuburgCoordinatorAustria
Links
Data: CORDIS, © European Union