Optimal Transport and Quantum Information Geometry Research Group

The central goal of our project is to study the geometry of classical and quantum structures which are endowed with transport-related metrics (like the Wasserstein metric) or entropic dissimilarity measures.

The main research objectives are the following:

• (G1) Classical Wasserstein spaces - Determining the structure of isometries of p-Wasserstein spaces built e.g. on spheres, projective spaces, Banach spaces, Hadamard spaces, non-branching spaces. The crucial novelty here is that we study non-Euclidean underlying geometries, and we consider not only the case p=2 (which is the most intensively studied so far) but all values of p>0.
• (G2) Quantum Wasserstein spaces - Developing the theory of quantum optimal transportation with respect to general (non-quadratic) cost operators. The main novelty is the application of a two-variable tensor product functional calculus. The other main objective of this part of the project is to describe the structure of quantum Wasserstein isometries of qubits and higher dimensional quantum state spaces.
• (G3) Metrics and other entropic dissimilarity measures on quantum states - Study of the geometry of the quantum state space equipped with general quantum Jensen divergences, with a special emphasis on the Riemannian metric tensor, geodesics, and barycenters. This research direction is absolutely new in quantum information geometry: even the triangle inequality for these distances is a very recent result published in 2021.

Publications 

[1] Z. M. Balogh, G. Kiss, T. Titkos, D. Virosztek. Isometric rigidity of the Wasserstein space over the plane with the maximum metric. Preprint, arXiv:2411.07051 (2024). Link(s): arXiv

[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 W_1(G) over Carnot groups. Preprint, arXiv:2305.05492 (2023). Link(s): arXiv

[4] Z. M. Balogh, T. Titkos, D. Virosztek. Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group. Preprint, arXiv:2303.15095 (2023). Link(s): arXiv

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

[2] 17th Workshop on Numerical Ranges and Numerical Radii, Taichung, Taiwan, invited talks, 19-21/06/2025 (T. Titkos and D. Virosztek — upcoming)

[3] Dynamics of Density Operators Workshop, IPAM, UCLA, Los Angeles, USA, invited talk, 28/04–02/05/2025 (D. Virosztek — upcoming)

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

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

2024

[6] 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)

[7] 35th International Workshop on Operator Theory and its Applications, University of Kent, Canterbury, UK, talks, 15/08/2024 (T. Titkos and D. Virosztek)

[8] 2nd Analysis Mathematica Conference, Renyi Institute, Budapest, Hungary, invited talk, 31/07/2024 (D. Virosztek)

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

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

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

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

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

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

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

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

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

[18] 51th Winter School in Abstract Analysis, Vlachovice, Czech Republic, talk, 15/01/2024 (D. Virosztek)

2023

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

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

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

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

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

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

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

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

[27] 50th Winter School in Abstract Analysis, Svratka, Czech Republic, talks, 11/01/2023 (T. Titkos and D. Virosztek)

2022

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

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

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

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

[32] 24th Conference of the International Linear Algebra Society (ILAS), Galway, Ireland, invited talks, 20-24/06/2022 (T. Titkos and D. Virosztek)

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

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

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

[36] 49th Winter School in Abstract Analysis, Svratka, Czech Republic, talks, 09/01/2022 (T. Titkos and D. Virosztek)

2021

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

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

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

Events hosted:

Optimal Transport on Quantum Structures semester - 2022. Sept-Dec