Optimal transport

The MTA–RI Momentum Optimal Transport and Quantum Information Geometry Research Group focuses on classical optimal transport, quantum information theory, and the developing field of quantum optimal transport theory, which arose from the synergistic relationship between the former two. Classical Wasserstein spaces are closely related to optimal transport theory and have proven to be extremely important in recent years in both mathematics (probability theory, metric geometry, financial mathematics) and applied sciences (image processing, machine learning). Quantum Wasserstein spaces and quantum optimal transport theory are still emerging, with many different and interesting formulations being explored and many open problems remaining in this area. The goal of the group is to determine the fundamental geometric properties of classical Wasserstein spaces and quantum state spaces equipped with entropic distances, and to participate in the construction of a non-quadratic quantum optimal transport theory.

Publications 

[1] Z. M. Balogh, G. Kiss, T. Titkos, D. Virosztek. Isometric rigidity of the Wasserstein space over the plane with the maximum metric. Canad. J. Math. (2025). Link(s): arXiv, online version

[2] Z. M. Balogh, E. Ströher, T. Titkos, D. Virosztek, Rigid and flexible Wasserstein spaces. Preprint, arXiv:2502.09364 (2025). Links: arXiv

[3] Z. M. Balogh, T. Titkos, D. Virosztek. Isometric rigidity of the Wasserstein space W₁(G) over Carnot groups. Potential Analysis (2025) (accepted for publication). Link(s): arXiv

[4] Z. M. Balogh, T. Titkos, D. Virosztek. Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group. Rev. Mat. Iberoam. 41 (2025), 2055-2084. Link(s): arXiv, published version

[5] G. Bunth. On Quantum Rényi Divergences. PhD thesis, BME, 2024. Link(s): online version

[6] G. Bunth, J. Pitrik, T. Titkos, D. Virosztek. On the metric property of quantum Wasserstein divergences. Physical Review A 110 (2024), 022211. Link(s): arXiv,  published version

[7] G. Bunth, J. Pitrik, T. Titkos, D. Virosztek. Wasserstein distances and divergences of order p by quantum channels. Preprint, arXiv:2501.08066 (2025). Links: arXiv

[8] M. Cúth, M. l. Doucha, T. Titkos. Isometries of Lipschitz-free Banach spaces. J. Lond. Math. Soc. 110 (2024), e70000. Link(s): arXiv, published version 

[9] G.P. Gehér, A. Hrušková, T. Titkos, D. Virosztek. Isometric rigidity of Wasserstein spaces over Euclidean spheres. J. Math. Anal. Appl. 542 (2025), 128810. Link(s): arXiv, published version 

[10] G.P. Gehér, J. Pitrik, T. Titkos, D. Virosztek. Quantum Wasserstein isometries on the qubit state space. J. Math. Anal. Appl. 522 (2023), 126955. Link(s): arXiv, published version

[11] G.P. Gehér, T. Titkos, D. Virosztek. Isometric rigidity of Wasserstein tori and spheres. Mathematika 69 (2023), 20-32. Link(s): arXiv, published version

[12] G.P. Gehér, T. Titkos, D. Virosztek. On isometries of Wasserstein spaces. RIMS Kôkyûroku Bessatsu B93 (2023), 239-250. Link(s): published version 

[13] G.P. Gehér, T. Titkos, D. Virosztek. On the exotic isometry flow of the quadratic Wasserstein space over the real line. Linear Algebra Appl. 693 (2024), 39-48. Link(s): published version 

[14] G.P. Gehér, T. Titkos, D. Virosztek, The isometry group of Wasserstein spaces: the Hilbertian case. J. London Math. Soc. 106 (2022), 3865-3894. Link(s): arXiv, published version  

[15] G. Kiss, T. Titkos. Isometric rigidity of Wasserstein spaces: the graph metric case. Proc. Amer. Math. Soc. 150 (2022), 4083-4097. Link(s): arXiv, published version

[16] J. Maas, S. Rademacher, T. Titkos, D. Virosztek (Eds.). Optimal transport on quantum structures. Bolyai Society Mathematical Studies, Volume 29 (2024). Link(s): published version

[17] M. Mosonyi, G. Bunth, and P. Vrana. Geometric relative entropies and barycentric Rényi divergences. Linear Algebra Appl. 699 (2024), 159–276. Link(s): arXiv, published version

[18] L. Portinale, S. Rademacher, D. Virosztek. Limit theorems for empirical measures of interacting quantum systems in Wasserstein space. Preprint arXiv:2312.00541 (2023). Link(s): arXiv

[19] A. Ramanantoanina, T. Titkos. Lattice properties of strength functions. Acta Sci. Math. (Szeged) 90 (2024), 679-689. Link(s): published version

[20] R. Simon, D. Virosztek. Preservers of the p-power and the Wasserstein means on 2×2 matrices.  Electron. J. Linear Algebra 39 (2023), 395-408. Link(s): arXiv, published version

[21] R. Simon, D. Virosztek, Isometries of the qubit state space with respect to quantum Wasserstein distances. Linear Algebra Appl. 714 (2025), 1-14. Link(s): arXiv, published version 

[22] G. Toth, J. Pitrik. Quantum Wasserstein distance based on an optimization over separable states. Quantum 7 (2023), 1143. Link(s): arXiv, published version

[23] D. Virosztek, Operator means, barycenters, and fixed point equations. Acta Sci. Math. (Szeged) 90 (2024), 391-408. Link(s): arXiv, published version 

Conference and seminar talks 

2025

[1] Appl.Anal. and MILAB Seminar, Eötvös Loránd University, talk 22/09/2025 (T. Titkos )

[2] 26th Conference of the International Linear Algebra Society (ILAS), Kaohsiung, Taiwan, invited talks, 23-27/06/2025 (T. Titkos and D. Virosztek)

[3] 17th Workshop on Numerical Ranges and Numerical Radii, Taichung, Taiwan, invited talks, 19-21/06/2025 (T. Titkos and D. Virosztek)

[4] Analysis Seminar, Renyi Institute, Budapest, Hungary, talk, 15/05/2025 (G. Bunth)

[5] Dynamics of Density Operators Workshop, IPAM, UCLA, Los Angeles, USA, invited talk, 28/04/2025 (D. Virosztek)

[6] Analysis Seminar at Politecnico di Milano, Dipartimento di Matematica, Milano, Italy, invited talk 27/03/2025 (T. Titkos )

[7] Analysis and Applied Mathematics Seminar, Bocconi University, Milano, Italy, invited talk, 04/03/2025 (T. Titkos)

[8] Analysis Seminar, Renyi Institute, Budapest, Hungary, talk, 29/05/2025 (J. Pitrik)

2024

[9] Quantum Optimal Transport: Theory and Applications (QOTTA) Workshop, Palazzone di Cortona, Italy, invited and contributed talks, 1-6/09/2024 (J. Pitrik, T. Titkos, D. Virosztek)

[10] 35th International Workshop on Operator Theory and its Applications, University of Kent, Canterbury, UK, talks, 15/08/2024 (T. Titkos and D. Virosztek)

[11] 2nd Analysis Mathematica Conference, Renyi Institute, Budapest, Hungary, invited talk, 31/07/2024 (D. Virosztek)

[12] 9th European Congress of Mathematics, Sevilla, Spain, talk, 16/07/2024 (T. Titkos)

[13]  Summer School in Mathematics, Eötvös Loránd University, invited lecture series, 17-21/06/2024 (T. Titkos and D. Virosztek) 

[14] Optimal transport minicourse at the Analysis Seminar of the Rényi Institute, three lectures, April-May of 2024 (T. Titkos and D. Virosztek)

[15] International Workshop on Matrix Analysis and Quantum Information Theory (MAA&QIT2024), Hue University, Hue City, Vietnam, invited (online) talk, 05/06/2024 (D. Virosztek)

[16] Submodularity seminar (org.: László Lovász et al.), Renyi Institute, Budapest, Hungary, survey talk, 30/04/2023 (T. Titkos)

[17] Discrete and Continuous Structures in Mathematics Workshop, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, invited talk, 26/04/2024 (D. Virosztek)

[18] Trojan Math Seminar, Troy University, Troy, USA, invited (online) talk, 07/03/2024 (D. Virosztek)

[19] Quantum Glue Meeting, UPV/EHU Seminar, Bilbao, On quantum Wasserstein Distance, 14/02/2024 (J. Pitrik)

[20] Bilbao Analsyis and PDE Seminar (BCAM), On quantum Wasserstein distance, 25/01/2024 (J. Pitrik)

[21] 51th Winter School in Abstract Analysis, Vlachovice, Czech Republic, talk, 15/01/2024 (D. Virosztek)

2023

[22] Analysis Seminar, Renyi Institute, Budapest, Hungary, talk, 30/11/2023 (D. Virosztek)

[23] Analysis Seminar, Renyi Institute, Budapest, Hungary, talk, 23/11/2023 (G. Bunth)

[24] Optimal transport: classical and quantum. Two invited lecture series at the Niigata University, Niigata, Japan, November of 2023 (D. Virosztek)

[25] Recent Progress in Operator Theory and its Applications Workshop, RIMS Kyoto, Japan, invited talk, 07/11/2023 (D. Virosztek)

[26] Submodularity seminar (org.: László Lovász et al.), Renyi Institute, Budapest, Hungary, survey talk, 10/10/2023 (D. Virosztek) 

[27] Æsy to define, Hard to analyse: First conference on Lipschitz-free spaces, Centre Diocésain Besançon, France, invited talk, 20/09/2023 (T. Titkos)

[28] Séminaire de Mathématiques et Colloquium, Institut de Recherche en Informatique, Mathématiques, Automatique et Signal, Mulhouse, France, invited talk, 18/09/2023 (T. Titkos) 

[29] International Workshop on Matrix Analysis and its Applications, Quy Nhon, Vietnam, talk, 08/07/2023 (J. Pitrik).

[30] Focused Workshop on Moffat's Magnetic Relaxation Problem (org.: László Székelyhidi Jr.), Erdos Center, Budapest, Hungary, invited talk, 05/07/2023 (D. Virosztek)

[31] 50th Winter School in Abstract Analysis, Svratka, Czech Republic, talks, 11/01/2023 (T. Titkos and D. Virosztek)

2022

[32] Real and Abstract Analysis Seminar, Charles University, Prague, Czech Republic, invited talk, 23/11/2022 (T. Titkos)

[33] "Kutszem" Research Seminar, Renyi Institute, Budapest, Hungary, invited seminar talk, 21/11/2022 (D. Virosztek) 

[34] Analysis Seminar, Eötvös Loránd University, Budapest, Hungary, invited seminar talk, 21/11/2022 (D. Virosztek) 

[35] AMS-SMF-EMS joint international meeting, Special Session on Quantitative Geometry of Transportation Metrics, Grenoble, France, invited talk, 18/07/2022 (D. Virosztek)

[36] 24th Conference of the International Linear Algebra Society (ILAS), Galway, Ireland, invited talks, 20-24/06/2022 (T. Titkos and D. Virosztek)

[37] Budapest Graduate Seminar, Renyi Institute, invited lectures, 20/04/2022 (T. Titkos and D. Virosztek)

[38] Mathematical Colloquium, University of Bern, Switzerland, invited seminar talk, 04/04/2022 (D. Virosztek)

[39] Institute Seminar, Institute for Solid State Physics and Optics, Wigner Research Center for Physics, 22/02/2022 (J. Pitrik)

[40] 49th Winter School in Abstract Analysis, Svratka, Czech Republic, talks, 09/01/2022 (T. Titkos and D. Virosztek)

2021

[41] Analysis Seminar, Renyi Institute, Budapest, Hungary, talk, 09/12/2021 (D. Virosztek)

[42] "Research on preserver problems on Banach algebras and related topics" Workshop at RIMS Kyoto, Japan, invited talk (online), 27/10/2021 (D. Virosztek)

[43] "Kutszem" Research Seminar, Renyi Institute, Budapest, Hungary, talk, 18/10/2021 (D. Virosztek)

Events hosted:

Optimal Transport on Quantum Structures semester - 2022. szeptember-december