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Approach to optimal quantum transport via states over time

Matt Hoogsteder-Riera, John Calsamiglia, and Andreas Winter

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AbstractWe approach the problem of constructing a quantum analogue of the immensely fruitful classical transport cost theory of Monge from a new angle. Going back to the original motivations, by which the transport is a bilinear function of a mass distribution (without loss of generality a probability density) and a transport plan (a stochastic kernel), we explore the quantum version where the mass distribution is generalised to a density matrix, and the transport plan to a completely positive and trace preserving map. These two data are naturally integrated into their Jordan product, which is called state over time (“stote''), and the transport cost is postulated to be a linear function of it. We explore the properties of this transport cost, as well as the optimal transport cost between two given states (simply the minimum cost over all suitable transport plans). After that, we analyse in considerable detail the case of unitary invariant cost, for which we can calculate many costs analytically. These findings suggest that our quantum transport cost is qualitatively different from Monge's classical transport.Popular summaryHow can one build fortifications with minimal material displacement? How can one optimally distribute resources produced in several production centers to multiple distributions centers? Classical optimal transport provides mathematical answers to such questions by assigning a cost to every possible transport plan and selecting the cheapest one. In this work, we explore how this mathematical framework can be generalized to quantum systems. We seek a quantum formulation that mirrors the structure of the classical theory as closely as possible. This leads naturally to the notion of a state over time (or stote), a mathematical object that combines a quantum state and the process acting on it into a single description, analogous to the role played by joint probability distributions in the classical theory. Using stotes, we define a broad family of quantum transport costs and study the corresponding optimization problem between quantum states. We investigate the mathematical properties of the resulting framework and analyze in detail a particularly symmetric setting that allows for closed form results. Our work lays the groundwork for future studies aimed at identifying physically meaningful notions of quantum transport.► BibTeX data@article{HoogstederRiera2026approachtooptimal, doi = {10.22331/q-2026-10-14-2236}, url = {https://doi.org/10.22331/q-2026-10-14-2236}, title = {Approach to optimal quantum transport via states over time}, author = {Hoogsteder-Riera, Matt and Calsamiglia, John and Winter, Andreas}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2236}, month = oct, year = {2026} }► References [1] Gaspard Monge. ``Mémoire sur la théorie des déblais et des remblais''. De l'Imprimerie Royale. Paris (1781). [2] Leonid V. Kantorovich. ``On a problem of Monge''. Uspekhi Mat. Nauk. 3, 225–226 (1948). https://doi.org/10.1007/s10958-006-0050-9 [3] Leonid V. Kantorovitch. ``On the Translocation of Masses''. Management Science 5, 1–4 (1958). https://doi.org/10.1287/mnsc.5.1.1 [4] Cédric Villani. ``Optimal transport: Old and new''. Volume 338 of Grundlehren der mathematischen Wissenschaften. Springer Verlag. Berlin, Paris, Dakar (2008). https://doi.org/10.1007/978-3-540-71050-9 [5] Karol Życzkowski and Wojciech Słomczyński. ``The Monge distance between quantum states''. Journal of Physics A: Mathematical and General 31, 9095–9104 (1998). https://doi.org/10.1088/0305-4470/31/45/009 [6] Karol Życzkowski and Wojciech Słomczyński. ``The Monge metric on the sphere and geometry of quantum states''. Journal of Physics A: Mathematical and General 34, 6689–6722 (2001). https://doi.org/10.1088/0305-4470/34/34/311 [7] Matt Hoogsteder-Riera. ``A transport approach to distances in quantum systems''. Bachelor's thesis. Universitat Autònoma de Barcelona, Department of Physics. (2018). [8] Giacomo De Palma and Dario Trevisan. ``Quantum Optimal Transport with Quantum Channels''.

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Computer Physics Communications 184, 1234–1240 (2013). https://doi.org/10.1016/j.cpc.2012.11.019Cited by[1] Seok Hyung Lie and James Fullwood, "Multipartite Quantum States over Time from Two Fundamental Assumptions", Physical Review Letters 135 23, 230204 (2025). [2] Gilles Barthe, Minbo Gao, Jam Kabeer Ali Khan, Matthijs Muis, Ivan Renison, Keiya Sakabe, Michael Walter, Yingte Xu, Tianshi Yu, and Li Zhou, "Complete Relational Logic for Infinite-Dimensional Quantum Programs with Unbounded Assertions", arXiv:2510.07051, (2025). [3] Seok Hyung Lie and Hyukjoon Kwon, "Probing Quantum States over Spacetime through Interferometry", Physical Review Letters 136 25, 250201 (2026). [4] Rocco Duvenhage, Dylan van Zyl, and Paola Zurlo, "Fermionic optimal transport", arXiv:2510.02888, (2025). The above citations are from SAO/NASA ADS (last updated successfully 2026-10-15 19:56:18). The list may be incomplete as not all publishers provide suitable and complete citation data.On Crossref's cited-by service no data on citing works was found (last attempt 2026-10-15 19:56:17).This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWe approach the problem of constructing a quantum analogue of the immensely fruitful classical transport cost theory of Monge from a new angle. Going back to the original motivations, by which the transport is a bilinear function of a mass distribution (without loss of generality a probability density) and a transport plan (a stochastic kernel), we explore the quantum version where the mass distribution is generalised to a density matrix, and the transport plan to a completely positive and trace preserving map. These two data are naturally integrated into their Jordan product, which is called state over time (“stote''), and the transport cost is postulated to be a linear function of it. We explore the properties of this transport cost, as well as the optimal transport cost between two given states (simply the minimum cost over all suitable transport plans). After that, we analyse in considerable detail the case of unitary invariant cost, for which we can calculate many costs analytically. These findings suggest that our quantum transport cost is qualitatively different from Monge's classical transport.Popular summaryHow can one build fortifications with minimal material displacement? How can one optimally distribute resources produced in several production centers to multiple distributions centers? Classical optimal transport provides mathematical answers to such questions by assigning a cost to every possible transport plan and selecting the cheapest one. In this work, we explore how this mathematical framework can be generalized to quantum systems. We seek a quantum formulation that mirrors the structure of the classical theory as closely as possible. This leads naturally to the notion of a state over time (or stote), a mathematical object that combines a quantum state and the process acting on it into a single description, analogous to the role played by joint probability distributions in the classical theory. Using stotes, we define a broad family of quantum transport costs and study the corresponding optimization problem between quantum states. We investigate the mathematical properties of the resulting framework and analyze in detail a particularly symmetric setting that allows for closed form results. Our work lays the groundwork for future studies aimed at identifying physically meaningful notions of quantum transport.► BibTeX data@article{HoogstederRiera2026approachtooptimal, doi = {10.22331/q-2026-10-14-2236}, url = {https://doi.org/10.22331/q-2026-10-14-2236}, title = {Approach to optimal quantum transport via states over time}, author = {Hoogsteder-Riera, Matt and Calsamiglia, John and Winter, Andreas}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2236}, month = oct, year = {2026} }► References [1] Gaspard Monge. ``Mémoire sur la théorie des déblais et des remblais''. De l'Imprimerie Royale. Paris (1781). [2] Leonid V. Kantorovich. ``On a problem of Monge''. Uspekhi Mat. Nauk. 3, 225–226 (1948). https://doi.org/10.1007/s10958-006-0050-9 [3] Leonid V. Kantorovitch. ``On the Translocation of Masses''. Management Science 5, 1–4 (1958). https://doi.org/10.1287/mnsc.5.1.1 [4] Cédric Villani. ``Optimal transport: Old and new''. Volume 338 of Grundlehren der mathematischen Wissenschaften. Springer Verlag. Berlin, Paris, Dakar (2008). https://doi.org/10.1007/978-3-540-71050-9 [5] Karol Życzkowski and Wojciech Słomczyński. ``The Monge distance between quantum states''. Journal of Physics A: Mathematical and General 31, 9095–9104 (1998). https://doi.org/10.1088/0305-4470/31/45/009 [6] Karol Życzkowski and Wojciech Słomczyński. ``The Monge metric on the sphere and geometry of quantum states''. Journal of Physics A: Mathematical and General 34, 6689–6722 (2001). https://doi.org/10.1088/0305-4470/34/34/311 [7] Matt Hoogsteder-Riera. ``A transport approach to distances in quantum systems''. Bachelor's thesis. Universitat Autònoma de Barcelona, Department of Physics. (2018). [8] Giacomo De Palma and Dario Trevisan. ``Quantum Optimal Transport with Quantum Channels''.

Springer Berlin Heidelberg. Berlin, Heidelberg (1983). https://doi.org/10.1007/3540127321_22 [35] Michael A. Nielsen and Isaac L. Chuang. ``Quantum computation and quantum information''.

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