Optimális transzport

Az MTA–RI Lendület Optimális Transzport és Kvantum Információ-geometria Kutatócsoport klasszikus optimális transzporttal, kvantum-információelmélettel, valamint az előbbiek szinergikus kapcsolatából született rendkívül friss tudományterülettel, a kvantum optimális transzport elméletével foglalkozik. A klasszikus Wasserstein-terek szorosan kapcsolódnak az optimális transzport elméletéhez, és az utóbbi években rendkívül fontosnak bizonyultak mind a matematikában (valószínűségelmélet, metrikus geometria, pénzügyi matematika), mind az alkalmazott tudományokban (képfeldolgozás, gépi tanulás). A kvantum Wasserstein-terek és a kvantum optimális transzport elmélete éppen kialakulóban van, rengeteg különböző és érdekes megfogalmazást vizsgálnak, rengeteg a nyitott probléma ezen a területen. A csoport célja klasszikus Wasserstein-terek és entrópikus távolságokkal ellátott kvantum állapotterek alapvető geometriai jellemzőinek meghatározása, illetve részvétel a nemkvadratikus kvantum optimális transzport elméletének felépítésében.

Publikációk

[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), DOI: 10.4153/S0008414X25101053. Links: arXiv, published version

[2] Z. M. Balogh, E. Ströher, T. Titkos, D. Virosztek. Wasserstein rigidity over R^n with smooth norms. Mathematika 72 (2026), e70132. Links: arXiv, published version

[3] Z. M. Balogh, E. Ströher, D. Virosztek. Restoring Wasserstein rigidity with a single point. J. Math. Anal. Appl. 564 (2026), 130890. Links: arXiv, published version

[4] Z. M. Balogh, T. Titkos, D. Virosztek. Isometric rigidity of the Wasserstein space W_1(G) over Carnot groups. Potential Analysis 64 (2026), article nr. 1. Links: arXiv, published version

[5] 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. Links: arXiv, published version

[6] G. Bunth. On Quantum Rényi Divergences. PhD thesis, BME, 2024. Link: published version

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

[8] G. Bunth, J. Pitrik, T. Titkos, D. Virosztek, Strong Kantorovich duality for quantum optimal transport with generic cost and optimal couplings on quantum bits. Linear Algebra Appl. (2026), DOI: 10.1016/j.laa.2026.04.028. Links: arXiv, published version

[9] G. Bunth, J. Pitrik, T. Titkos, D. Virosztek, The swap transpose on couplings translates to Petz’ recovery map on quantum channels. Preprint, arXiv:2512.04919 (2025). Link: arXiv

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

[11] G. Bunth, E. Szabó, D. Virosztek. Quantum Wasserstein isometries of the n-qubit state space: a Wigner-type result. Proc. Amer. Math. Soc. (2026), DOI: 10.1090/proc/17919. Links: arXiv, published version

[12] G. Bunth, Wigner symmetries single out symmetric Wasserstein distances in all finite dimensions. Preprint, arXiv:2607.08298 (2026). Link: arXiv

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

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

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

[16] G.P. Gehér, J. Pitrik, T. Titkos, D. Virosztek. Transport distances and their preservers on quantum states. Linear Algebra Appl. (2026), DOI:10.1016/j.laa.2026.10.012. Link: published version 

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

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

[19] 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: published version 

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

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

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

[23] B. T. Morvay, Sz. Torma, J. Pitrik, L. Szegletes. Enhancing multi-paradigm EEG signal classification in cross-subject settings using optimal transport. Biomed. Signal Proc. Control 113 (2026), 108892. Link: published version 

[24] B.T. Morvay, S. Torma, B. Beres, L. Szegletes, J. Pitrik. An optimal transport-based domain adaptation method for enhancing cross-subject EEG emotion recognition. (2025) Proceedings of 2025 IEEE 16th International Conference on Cognitive Infocommunications (CogInfoCom)

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

[26] L. Portinale, S. Rademacher, D. Virosztek. Limit theorems for empirical measures of interacting quantum systems in Wasserstein space. Advances in Mathematics 495 (2026), 110972. Links: arXiv, published version

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

[28] 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. Links: arXiv, published version

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

[30] E. Szabó, D. Virosztek, Quantum Wasserstein isometries with a single observable generating the transport cost. Acta Sci. Math. (Szeged) (2026), DOI: 10.1007/s44146-026-00254-5. Link: published version

[31] T. Titkos, A negative example for a positive-sphere variant of Tingley’s problem, submitted manuscript

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

[33] G. Tóth, J. Pitrik. General method for obtaining the energy minimum of spin Hamiltonians for separable states. Preprint, arXiv:2605.03022 (2026). Link: arXiv

[34] G. Tóth, J. Pitrik. Relations between different definitions of the quantum Wasserstein distance.  Preprint, arXiv:2605.03027 (2026). Link: arXiv

[35]  G. Tóth, J. Pitrik Quantum Wasserstein distance and its relation to several types of fidelities. Preprint, arXiv:2506.14523 (2025). Link: arXiv

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

Konferencia és szeminárium előadások

2026

[1] Quantum Information and Chaos Seminar, Jagiellonian University, Cracow, Poland, invited talk, 02/11/2026 (D. Virosztek, upcoming)

[2] Quantum Information Theory Seminar, Budapest University of Technology and Economics, Budapest, Hungary, talk, 02/10/2026 (G. Bunth)

[3] Workshop on Preserver Problems in Japan (PPJP2026), RIMS Kyoto, Japan, talks, 24-28/08/2026 (G. Bunth, J. Pitrik, and T. Titkos)

[4] 37th International Workshop on Operator Theory and its Applications, Laval University, Quebec City, Canada, invited talks, 3-7/08/2026 (G. Bunth, J. Pitrik, T. Titkos, and D. Virosztek)

[5] International Congress of Mathematicians 2026, Philadelphia, USA, contributed talks, 23-30/07/2026 (T. Titkos and D. Virosztek)

2025

[6] IMUS Seminar, Instituto Matemáticas de la Universidad de Sevilla, Sevilla, Spain, invited talk, 17/10/2025 (T.Titkos)

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

[8] 26th Conference of the International Linear Algebra Society (ILAS), Kaohsiung, Taiwan, invited talks, 23-27/06/2025 (T. Titkos and D. Virosztek)

[9] 17th Workshop on Numerical Ranges and Numerical Radii, Taichung, Taiwan, invited talks, 19-21/06/2025 (T. Titkos and D. Virosztek)

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

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

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

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

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

2024

[15] 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)

[16] 35th International Workshop on Operator Theory and its Applications, University of Kent, Canterbury, UK, talks, 15/08/2024 (T. Titkos and D. Virosztek)

[17] 2nd Analysis Mathematica Conference, Renyi Institute, Budapest, Hungary, invited talk, 31/07/2024 (D. Virosztek)

[18] 9th European Congress of Mathematics, Sevilla, Spain, talk, 16/07/2024 (T. Titkos)

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

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

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

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

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

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

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

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

[27] 51th Winter School in Abstract Analysis, Vlachovice, Czech Republic, talk, 15/01/2024 (D. Virosztek)

2023

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

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

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

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

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

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

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

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

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

[37] 50th Winter School in Abstract Analysis, Svratka, Czech Republic, talks, 11/01/2023 (T. Titkos and D. Virosztek)

2022

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

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

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

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

[42] 24th Conference of the International Linear Algebra Society (ILAS), Galway, Ireland, invited talks, 20-24/06/2022 (T. Titkos and D. Virosztek)

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

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

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

[46] 49th Winter School in Abstract Analysis, Svratka, Czech Republic, talks, 09/01/2022 (T. Titkos and D. Virosztek)

2021

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

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

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

Események

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

Sajtómegjelenések

Magyar Katolikus Rádió Kerengő című műsorában elhangzott interjú - 2021. október 19.

MTA Lendületesek  - interjú Virosztek Dániellel - 2023. január 11.