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Peer-reviewedCC BY 4.0Quantum

Approach to optimal quantum transport via states over time

  • Matt Hoogsteder-Riera1
  • John Calsamiglia1
  • Andreas Winter1,2,3,4
4 affiliations
  1. 1Grup d'Informació Quàntica, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (BCN), Spain
  2. 2ICREA—Institució Catalana de Recerca i Estudis Avançats, Pg. Lluís Companys, 23, 08010 Barcelona, Spain
  3. 3Department Mathematik/Informatik—Abteilung Informatik, Universität zu Köln, Albertus-Magnus-Platz, 50923 Köln, Germany
  4. 4Institute for Advanced Study, Technische Universität München, Lichtenbergstraße 2a, 85748 Garching, Germany
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Abstract

We 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.

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Cite

BibTeXRIS
BibTeX
@article{hoogstederriera2026approach,
  title = {{Approach to optimal quantum transport via states over time}},
  author = {Matt Hoogsteder-Riera and John Calsamiglia and Andreas Winter},
  journal = {Quantum},
  year = {2026},
  month = oct,
  doi = {10.22331/q-2026-10-14-2236},
  url = {https://quantum-journal.org/papers/q-2026-10-14-2236/},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Approach to optimal quantum transport via states over time
AU  - Matt Hoogsteder-Riera
AU  - John Calsamiglia
AU  - Andreas Winter
PY  - 2026
DA  - 2026/10/14
JO  - Quantum
DO  - 10.22331/q-2026-10-14-2236
UR  - https://quantum-journal.org/papers/q-2026-10-14-2236/
AB  - We 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.
SN  - 2521-327X
ER  -