pith. sign in

arxiv: 2506.09794 · v1 · submitted 2025-06-11 · 🪐 quant-ph

Wasserstein Distances on Quantum Structures: an Overview

Emily Beatty This is my paper

Pith reviewed 2026-05-19 09:26 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum Wasserstein distancequantum optimal transportquantum informationWasserstein distancequantum statesoptimal transport reviewopen problems
0
0 comments X

The pith

Scattered proposals for Wasserstein distances on quantum states are compiled into one overview with no agreed single definition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gathers existing attempts to extend Wasserstein distances from classical probability measures to quantum states. It shows that these attempts remain disconnected across the literature and lack any shared agreement on which version qualifies as the correct quantum generalization. A reader would care because the classical versions already support results in machine learning, Markov chains, and concentration inequalities, so a clear map of their quantum counterparts could speed up similar advances for quantum systems. The review also flags open questions to steer further work in both optimal transport and quantum information.

Core claim

The literature on quantum Wasserstein distances remains scattered with very few links between different works and no consensus on a true quantum version. This review collects the proposals under one framework, surveys the state of the art including their use in functional inequalities, many-body convergence, and quantum generative models, and outlines open problems together with possible future directions for researchers entering from either classical optimal transport or quantum information.

What carries the argument

A comparative review framework that places multiple proposed quantum Wasserstein distances side by side to reveal relations, differences, and shared features.

If this is right

  • Functional inequalities previously known for classical measures extend to quantum states via these distances.
  • Convergence rates for solutions of many-body quantum equations become quantifiable.
  • Training of quantum generative adversarial networks improves through the new distance measures.
  • Open problems identified in the review become targets for connecting classical and quantum optimal transport.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The overview could help classical optimal transport researchers identify which quantum definitions preserve the most familiar properties such as triangle inequality or geodesic structure.
  • Applications in economics or concentration of measure might translate to quantum settings once the distances are placed in a common language.
  • Practical tests on small quantum devices could rank the proposals by computational cost and stability.

Load-bearing premise

The various existing proposals for quantum Wasserstein distances share enough common structure to be usefully compared and organized within a single review framework.

What would settle it

An exhaustive mapping of all published definitions that finds no overlapping properties, no shared axioms, and no workable relations between any pair would show the proposals cannot be organized together.

read the original abstract

The theory of optimal transport of probability measures has wide-ranging applications across a number of different fields, including concentration of measure, machine learning, Markov chains, and economics. The generalisation of optimal transport tools from probability measures to quantum states has shown great promise over the last few years, particularly in the development of the theory of Wasserstein-style distances and divergences between quantum states. Such distances have already led to a broad range of developments in the quantum setting such as functional inequalities, convergence of solutions in many-body physics, improvements to quantum generative adversarial networks, and more. However, the literature in this field is quite scattered, with very few links between different works and no real consensus on a `true' quantum Wasserstein distance. The aim of this review is to bring these works together under one roof and give a full overview of the state of the art in the development of quantum Wasserstein distances. We also present a variety of open problems and unexplored avenues in the field, and examine the future directions of this promising line of research. This review is written for those interested in quantum optimal transport in coming from both the fields of classical optimal transport and of quantum information theory, and as a resource for those working in one area of quantum optimal transport interested in how existing work may relate to their own.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. This review collects and organizes existing proposals for Wasserstein-style distances between quantum states, surveys their applications in functional inequalities, many-body convergence, quantum GANs and related areas, and identifies open problems and future research directions. It is written for readers from classical optimal transport and quantum information theory, with the explicit goal of relating scattered works that currently lack consensus on a canonical quantum Wasserstein distance.

Significance. If the survey accurately represents the cited literature, the manuscript would be a useful consolidation of a fragmented subfield. By explicitly linking proposals that have developed in isolation, it could reduce duplication of effort and surface cross-community connections; the inclusion of open problems further positions the work as a resource rather than a mere bibliography.

minor comments (3)
  1. The abstract states that the literature has 'very few links between different works'; the introduction should include a short table or diagram that explicitly maps which proposals are compared in later sections and which remain isolated, to make the claimed unifying contribution concrete.
  2. Notation for the various quantum Wasserstein distances (e.g., W_p, D_p, etc.) should be introduced in a single dedicated subsection with a comparison table of their defining properties (metric axioms satisfied, computational tractability, relation to classical OT) rather than piecemeal in each subsection.
  3. Several application paragraphs mention 'improvements to quantum generative adversarial networks' without citing the specific papers that use a given quantum Wasserstein distance; adding those references would strengthen the claim that the distances have already produced concrete developments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition that it consolidates a fragmented subfield and identifies open problems. We appreciate the recommendation for minor revision and will make appropriate updates to improve the presentation and accuracy of the review.

Circularity Check

0 steps flagged

Review paper with no derivations or self-referential reductions

full rationale

This is a survey whose stated purpose is to collect and relate existing proposals for quantum Wasserstein distances. No new equations, predictions, or quantitative claims are advanced that could reduce to fitted parameters, self-definitions, or self-citation chains. The precondition that the proposals share enough structure for comparison is the natural precondition for any review and does not constitute a circular step. The paper is self-contained against external benchmarks as a literature overview.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper; it introduces no new free parameters, axioms, or invented entities and instead surveys results from the existing literature.

pith-pipeline@v0.9.0 · 5748 in / 839 out tokens · 34505 ms · 2026-05-19T09:26:57.442629+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Holography and Optimal Transport: Emergent Wasserstein Spacetime in Harmonic Oscillator, SYK and Krylov Complexity

    hep-th 2026-04 unverdicted novelty 7.0

    Holographic spacetime emerges as the 1-Wasserstein space of quantum state distributions under optimal transport, matching AdS2 black hole geometry in the SYK model and identified with generalized Krylov complexity.

  2. Strong Kantorovich duality for quantum optimal transport with generic cost and optimal couplings on quantum bits

    math-ph 2025-10 unverdicted novelty 6.0

    Establishes Kantorovich duality for linearized non-quadratic quantum optimal transport realized by channels, determines optimal primal-dual solutions for qubits under state restrictions, and proves the triangle inequa...

  3. Relations between different definitions of the quantum Wasserstein distance for qubits

    quant-ph 2026-05 unverdicted novelty 5.0

    Two quantum Wasserstein distance definitions coincide for qubits with single-operator cost functions, implying the self-distance equals the Wigner-Yanase skew information.

Reference graph

Works this paper leans on

147 extracted references · 147 canonical work pages · cited by 3 Pith papers · 2 internal anchors

  1. [1]

    Bosonic coding: introduction and use cases,

    Victor V. Albert. “Bosonic coding: introduction and use cases”. In: arXiv:2211.05714 (Nov. 2022)

    work page arXiv 2022

  2. [2]

    On the detailed balance condition for non-hamiltonian systems

    Robert Alicki. “On the detailed balance condition for non-hamiltonian systems”. In: Reports on Mathematical Physics 10.2 (1976), pp. 249–258. issn: 0034-4877. doi: https://doi.org/10.1016/0034-4877(76)90046- X. url: https://www.sciencedirect.com/science/article/pii/003448777690046X

    work page doi:10.1016/0034-4877(76)90046- 1976

  3. [3]

    On the rate of convergence for the mean field approx- imation of Bosonic many-body quantum dynamics

    Zied Ammari, Marco Falconi, and Boris Pawilowski. “On the rate of convergence for the mean field approx- imation of Bosonic many-body quantum dynamics”. In: Communications in Mathematical Sciences 14.5 (2016), pp. 1417–1442. issn: 1945-0796. doi: 10.4310/cms.2016.v14.n5.a9 . url: http://dx.doi.org/ 10.4310/CMS.2016.v14.n5.a9. 34

    work page doi:10.4310/cms.2016.v14.n5.a9 2016

  4. [4]

    Schweidel

    Gil Appel, Juliana Neelbauer, and David A. Schweidel. Generative AI Has an Intellectual Property Problem

    work page

  5. [5]

    org / 2023 / 04 / generative - ai - has - an - intellectual - property - problem (visited on 05/22/2025)

    url: https : / / hbr . org / 2023 / 04 / generative - ai - has - an - intellectual - property - problem (visited on 05/22/2025)

    work page 2023

  6. [6]

    Resource-Dependent Complexity of Quantum Channels

    Roy Araiza et al. Resource-Dependent Complexity of Quantum Channels. 2025. arXiv: 2303.11304 [quant-ph]. url: https://arxiv.org/abs/2303.11304

    work page arXiv 2025

  7. [7]

    Wasserstein Generative Adversarial Networks

    Martin Arjovsky, Soumith Chintala, and L´ eon Bottou. “Wasserstein Generative Adversarial Networks”. In: Proceedings of the 34th International Conference on Machine Learning. Ed. by Doina Precup and Yee Whye Teh. Vol. 70. Proceedings of Machine Learning Research. PMLR, Aug. 2017, pp. 214–223. url: https : //proceedings.mlr.press/v70/arjovsky17a.html

    work page 2017

  8. [8]

    The mean field Schr¨ odinger problem: ergodic behavior, entropy estimates and functional inequalities

    Julio Backhoff et al. “The mean field Schr¨ odinger problem: ergodic behavior, entropy estimates and functional inequalities”. In: Probability Theory and Related Fields178 (Oct. 2020). doi: 10.1007/s00440-020-00977-8

    work page doi:10.1007/s00440-020-00977-8 2020

  9. [9]

    Weak coupling limit of the N-particle Schr¨ odinger equation

    Claude Bardos, Francois Golse, and Norbert Mauser. “Weak coupling limit of the N-particle Schr¨ odinger equation”. In: Methods and Applications of Analysis 7 (Jan. 2000). doi: 10.4310/MAA.2000.v7.n2.a2

    work page doi:10.4310/maa.2000.v7.n2.a2 2000

  10. [10]

    & Plenio, M

    T. Baumgratz, M. Cramer, and M. B. Plenio. “Quantifying Coherence”. In: Phys. Rev. Lett. 113 (14 Sept. 2014), p. 140401. doi: 10.1103/PhysRevLett.113.140401 . url: https://link.aps.org/doi/10.1103/ PhysRevLett.113.140401

    work page doi:10.1103/physrevlett.113.140401 2014

  11. [12]

    Quantum Statistical Learning via Quantum Wasserstein Natural Gradient

    Simon Becker and Wuchen Li. “Quantum Statistical Learning via Quantum Wasserstein Natural Gradient”. In: Journal of Statistical Physics 182.1 (Jan. 2021). issn: 1572-9613. doi: 10.1007/s10955-020-02682-1 . url: http://dx.doi.org/10.1007/s10955-020-02682-1

    work page doi:10.1007/s10955-020-02682-1 2021

  12. [13]

    A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

    Jean-David Benamou and Yann Brenier. “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem”. In: Numerische Mathematik 84.3 (2000), pp. 375–393

    work page 2000

  13. [14]

    Deviation bounds and concentration inequalities for quantum noises

    Tristan Benoist, Lisa H¨ anggli, and Cambyse Rouz´ e. “Deviation bounds and concentration inequalities for quantum noises”. In: Quantum 6 (Aug. 2022), p. 772. issn: 2521-327X. doi: 10.22331/q-2022-08-04-772 . url: http://dx.doi.org/10.22331/q-2022-08-04-772

    work page doi:10.22331/q-2022-08-04-772 2022

  14. [15]

    Integrable geodesic flows on surfaces

    Philippe Biane and Dan Voiculescu. “A Free Probability Analogue of the Wasserstein Metric on the Trace- State Space”. In: Geometric And Functional Analysis 11 (Dec. 2001), pp. 1125–1138. doi: 10.1007/s00039- 001-8226-4

    work page doi:10.1007/s00039- 2001

  15. [16]

    Effect of Rare Fluctuations on the Thermalization of Isolated Quantum Systems

    Giulio Biroli, Corinna Kollath, and Andreas M. L¨ auchli. “Effect of Rare Fluctuations on the Thermalization of Isolated Quantum Systems”. In: Physical Review Letters 105.25 (Dec. 2010). issn: 1079-7114. doi: 10. 1103/physrevlett.105.250401. url: http://dx.doi.org/10.1103/PhysRevLett.105.250401

    work page doi:10.1103/physrevlett.105.250401 2010

  16. [17]

    Maggiore, J

    R Bistro´ n, M Eckstein, and K ˙Zyczkowski. “Monotonicity of a quantum 2-Wasserstein distance”. In: Journal of Physics A: Mathematical and Theoretical 56.9 (Feb. 2023), p. 095301.issn: 1751-8121. doi: 10.1088/1751- 8121/acb9c8. url: http://dx.doi.org/10.1088/1751-8121/acb9c8

    work page doi:10.1088/1751- 2023

  17. [18]

    Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

    Bruce Blackadar. Operator Algebras: Theory of C*-Algebras and von Neumann Algebras . Springer, 2006. isbn: 9783540284864

    work page 2006

  18. [19]

    Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities

    S.G Bobkov and F G¨ otze. “Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities”. In: Journal of Functional Analysis 163.1 (1999), pp. 1–28

    work page 1999

  19. [20]

    AI, Copyright, and the Law: The Ongoing Battle Over Intellectual Property Rights

    Negar Bondari. AI, Copyright, and the Law: The Ongoing Battle Over Intellectual Property Rights . Feb. 2025

    work page 2025

  20. [21]

    Exponential Decay of Correlations Implies Area Law

    Fernando G. S. L. Brand˜ ao and Micha l Horodecki. “Exponential Decay of Correlations Implies Area Law”. In: Communications in Mathematical Physics 333.2 (Nov. 2014), pp. 761–798. issn: 1432-0916. doi: 10 . 1007/s00220-014-2213-8 . url: http://dx.doi.org/10.1007/s00220-014-2213-8

    work page doi:10.1007/s00220-014-2213-8 2014

  21. [22]

    Resource Theory of Quantum States Out of Thermal Equilibrium

    Fernando G. S. L. Brand˜ ao et al. “Resource Theory of Quantum States Out of Thermal Equilibrium”. In: Phys. Rev. Lett. 111 (25 Dec. 2013), p. 250404. doi: 10 . 1103 / PhysRevLett . 111 . 250404. url: https : //link.aps.org/doi/10.1103/PhysRevLett.111.250404

    work page doi:10.1103/physrevlett.111.250404 2013

  22. [23]

    Complete logarithmic Sobolev inequalities via Ricci curvature bounded below

    Michael Brannan, Li Gao, and Marius Junge. “Complete logarithmic Sobolev inequalities via Ricci curvature bounded below”. In: Advances in Mathematics 394 (2022), p. 108129. issn: 0001-8708. doi: https://doi. org/10.1016/j.aim.2021.108129 . url: https://www.sciencedirect.com/science/article/pii/ S0001870821005685. 35

    work page doi:10.1016/j.aim.2021.108129 2022

  23. [24]

    Path coupling: A technique for proving rapid mixing in Markov chains

    R. Bubley and M. Dyer. “Path coupling: A technique for proving rapid mixing in Markov chains”. In: Proceedings of the 38th Annual Symposium on Foundations of Computer Science . FOCS ’97. USA: IEEE Computer Society, 1997, p. 223. isbn: 0818681977

    work page 1997

  24. [25]

    Metric property of quantum wasserstein diver gences

    Gergely Bunth et al. “Metric property of quantum Wasserstein divergences”. In: Phys. Rev. A 110 (2 Aug. 2024), p. 022211. doi: 10.1103/PhysRevA.110.022211 . url: https://link.aps.org/doi/10.1103/ PhysRevA.110.022211

    work page doi:10.1103/physreva.110.022211 2024

  25. [26]

    A Mass Transportation Model for the Optimal Planning of an Urban Region

    Giuseppe Buttazzo and Filippo Santambrogio. “A Mass Transportation Model for the Optimal Planning of an Urban Region”. In: SIAM Review 51.3 (2009), pp. 593–610

    work page 2009

  26. [27]

    T owards optimal transport for quantum densities, 2021

    Emanuele Caglioti, Francois Golse, and Thierry Paul. “Towards Optimal Transport for Quantum Densities”. In: arXiv:2101.03256 (Jan. 2021)

    work page arXiv 2021

  27. [28]

    Rouz´ e, and D

    ’Angela Capel, Cambyse Rouz’e, and Daniel Stilck Francca. “The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions”. In: arXiv:2009.11817 (2020). url: https://api.semanticscholar.org/CorpusID:221879260

    work page arXiv 2009

  28. [29]

    Quantum optimal transport with convex regularization

    Emanuele Caputo et al. “Quantum optimal transport with convex regularization”. In: arXiv:2409.03698 (Sept. 2024)

    work page arXiv 2024

  29. [30]

    An Analog of the 2-Wasserstein Metric in Non-commutative Probability under which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy

    Eric A. Carlen and Jan Maas. “An Analog of the 2-Wasserstein Metric in Non-commutative Probability under which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy”. In: Communications in Mathematical Physics 331 (2014), pp. 887–926

    work page 2014

  30. [31]

    Carlen and Jan Maas

    Eric A. Carlen and Jan Maas. “Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance”. In: Journal of Functional Analysis 273.5 (2017), pp. 1810–1869. issn: 0022-1236. doi: https://doi.org/10.1016/j.jfa.2017.05.003 . url: https://www.sciencedirect.com/science/ article/pii/S0022123617301878

    work page doi:10.1016/j.jfa.2017.05.003 2017

  31. [32]

    Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems

    Eric A. Carlen and Jan Maas. “Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems”. In: Journal of Statistical Physics 178.2 (Nov. 2019), pp. 319–378. issn: 1572-

    work page 2019

  32. [33]

    Carlen and Jan Maas

    doi: 10.1007/s10955-019-02434-w . url: http://dx.doi.org/10.1007/s10955-019-02434-w

    work page doi:10.1007/s10955-019-02434-w

  33. [34]

    Information-theoretic generalization bounds for learning from quantum data

    Matthias C. Caro et al. “Information-theoretic generalization bounds for learning from quantum data”. In: Proceedings of Thirty Seventh Conference on Learning Theory . Ed. by Shipra Agrawal and Aaron Roth. Vol. 247. Proceedings of Machine Learning Research. PMLR, July 2024, pp. 775–839. url: https: //proceedings.mlr.press/v247/caro24a.html

    work page 2024

  34. [35]

    Quantum wasserstein GANs

    Shouvanik Chakrabarti et al. “Quantum wasserstein GANs”. In: Proceedings of the 33rd International Con- ference on Neural Information Processing Systems . Red Hook, NY, USA: Curran Associates Inc., 2019

    work page 2019

  35. [36]

    On Quantum Optimal Transport

    Sam Cole et al. “On Quantum Optimal Transport”. In: Mathematical Physics, Analysis and Geometry 26.2 (June 2023). issn: 1572-9656. doi: 10.1007/s11040-023-09456-7 . url: http://dx.doi.org/10.1007/ s11040-023-09456-7

    work page doi:10.1007/s11040-023-09456-7 2023

  36. [37]

    Noncommutative geometry

    Alain Connes. Noncommutative geometry. Academic Press, 1994. isbn: 9780121858605

    work page 1994

  37. [38]

    Quantum generative adversarial networks

    Pierre-Luc Dallaire-Demers and Nathan Killoran. “Quantum generative adversarial networks”. In: Physical Review A 98.1 (July 2018). issn: 2469-9934. doi: 10.1103/physreva.98.012324 . url: http://dx.doi. org/10.1103/PhysRevA.98.012324

    work page doi:10.1103/physreva.98.012324 2018

  38. [39]

    Gibbs sampling

    Alexander M. Dalzell et al. “Gibbs sampling”. In: Quantum Algorithms: A Survey of Applications and End- to-end Complexities. Cambridge University Press, 2025, pp. 243–249

    work page 2025

  39. [40]

    Classical shadows meet quantum optimal mass transport

    Giacomo De Palma, Tristan Klein, and Davide Pastorello. “Classical shadows meet quantum optimal mass transport”. In: Journal of Mathematical Physics 65.9 (Sept. 2024), p. 092201. issn: 0022-2488. doi: 10 . 1063/5.0178897. url: https://doi.org/10.1063/5.0178897

    work page doi:10.1063/5.0178897 2024

  40. [41]

    Quantum Concentration Inequalities

    Giacomo De Palma and Cambyse Rouz´ e. “Quantum Concentration Inequalities”. In:Annales Henri Poincar´ e 23.9 (Apr. 2022), pp. 3391–3429. issn: 1424-0661. doi: 10 . 1007 / s00023 - 022 - 01181 - 1. url: http : //dx.doi.org/10.1007/s00023-022-01181-1

    work page doi:10.1007/s00023-022-01181-1 2022

  41. [42]

    Quantum Optimal Transport with Quantum Channels

    Giacomo De Palma and Dario Trevisan. “Quantum Optimal Transport with Quantum Channels”. In: Annales Henri Poincar´ e22.10 (Mar. 2021), pp. 3199–3234. issn: 1424-0661. doi: 10.1007/s00023- 021- 01042- 3. url: http://dx.doi.org/10.1007/s00023-021-01042-3

    work page doi:10.1007/s00023- 2021

  42. [43]

    Quantum Optimal Transport: Quantum Channels and Qubits

    Giacomo De Palma and Dario Trevisan. “Quantum Optimal Transport: Quantum Channels and Qubits”. In: Optimal Transport on Quantum Structures . Springer Nature Switzerland, 2024, pp. 203–239. 36

    work page 2024

  43. [44]

    De Palma and D

    Giacomo De Palma and Dario Trevisan. “The Wasserstein Distance of Order 1 for Quantum Spin Systems on Infinite Lattices”. In: Annales Henri Poincar´ e24.12 (June 2023), pp. 4237–4282. issn: 1424-0661. doi: 10.1007/s00023-023-01340-y . url: http://dx.doi.org/10.1007/s00023-023-01340-y

    work page doi:10.1007/s00023-023-01340-y 2023

  44. [45]

    Limitations of Variational Quantum Algorithms: A Quantum Optimal Transport Approach

    Giacomo De Palma et al. “Limitations of Variational Quantum Algorithms: A Quantum Optimal Transport Approach”. In: PRX Quantum 4 (1 Jan. 2023), p. 010309. doi: 10 . 1103 / PRXQuantum . 4 . 010309. url: https://link.aps.org/doi/10.1103/PRXQuantum.4.010309

    work page doi:10.1103/prxquantum.4.010309 2023

  45. [46]

    DOI: 10.1109/TIT.2021.3076442

    Giacomo De Palma et al. “The Quantum Wasserstein Distance of Order 1”. In: IEEE Transactions on Information Theory 67.10 (2021), pp. 6627–6643. doi: 10.1109/TIT.2021.3076442

    work page doi:10.1109/tit.2021.3076442 2021

  46. [47]

    Rapid solution of problems by quantum computation

    David Deutsch and Richard Josza. “Rapid solution of problems by quantum computation”. In: Proceedings of the Royal Society A (Dec. 1992)

    work page 1992

  47. [48]

    Quantum statistical me- chanics in a closed system

    J. M. Deutsch. “Quantum statistical mechanics in a closed system”. In: Phys. Rev. A 43 (4 Feb. 1991), pp. 2046–2049. doi: 10.1103/PhysRevA.43.2046. url: https://link.aps.org/doi/10.1103/PhysRevA. 43.2046

    work page doi:10.1103/physreva.43.2046 1991

  48. [49]

    Lower Bound for Simulation Cost of Open Quantum Systems: Lipschitz Continuity Approach

    Zhiyan Ding et al. “Lower Bound for Simulation Cost of Open Quantum Systems: Lipschitz Continuity Approach”. In: Communications in Mathematical Physics 406.3 (Feb. 2025). issn: 1432-0916. doi: 10.1007/ s00220-025-05240-6 . url: http://dx.doi.org/10.1007/s00220-025-05240-6

    work page doi:10.1007/s00220-025-05240-6 2025

  49. [50]

    Quadratic Wasserstein metrics for vo n Neumann algebras via transport plans

    Rocco Duvenhage. “Quadratic Wasserstein metrics for von Neumann algebras via transport plans”. In: arXiv:2012.03564 (Dec. 2020)

    work page arXiv 2012

  50. [51]

    Wasserstein distance between noncommutative dynamical systems

    Rocco Duvenhage. “Wasserstein distance between noncommutative dynamical systems”. In: Journal of Math- ematical Analysis and Applications 527.1, Part 2 (2023), p. 127353. issn: 0022-247X. doi: https://doi. org/10.1016/j.jmaa.2023.127353 . url: https://www.sciencedirect.com/science/article/pii/ S0022247X23003566

    work page doi:10.1016/j.jmaa.2023.127353 2023

  51. [52]

    Quantum Wasserstein distance of order 1 between channels

    Rocco Duvenhage and Mathumo Mapaya. “Quantum Wasserstein distance of order 1 between channels”. In: Infinite Dimensional Analysis, Quantum Probability and Related Topics 26 (Apr. 2023). doi: 10.1142/ S0219025723500066

    work page 2023

  52. [53]

    Extending quantum detailed balance through optimal transport

    Rocco Duvenhage, Samuel Skosana, and Machiel Snyman. “Extending quantum detailed balance through optimal transport”. In: arXiv:2206.15287 (June 2022)

    work page arXiv 2022

  53. [54]

    Poincar´ e, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature

    Matthias Erbar and Max Fathi. “Poincar´ e, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature”. In: Journal of Functional Analysis 274.11 (2018), pp. 3056–3089. issn: 0022-1236. doi: https : / / doi . org / 10 . 1016 / j . jfa . 2018 . 03 . 011. url: https : //www.sciencedirect.com/science/artic...

    work page 2018

  54. [55]

    Ricci Curvature of Finite Markov Chains via Convexity of the Entropy

    Matthias Erbar and Jan Maas. “Ricci Curvature of Finite Markov Chains via Convexity of the Entropy”. In: Archive for Rational Mechanics and Analysis 206.3 (Aug. 2012), pp. 997–1038. issn: 1432-0673. doi: 10.1007/s00205-012-0554-z . url: http://dx.doi.org/10.1007/s00205-012-0554-z

    work page doi:10.1007/s00205-012-0554-z 2012

  55. [56]

    International Journal of The- oretical Physics 21(6/7), 467–488 (1982) https://doi.org/10.1007/BF02650179

    R.P Feynman. “Simulating physics with computers”. In: Int J Theor Phys 21 (1982), pp. 467–488. url: https://doi.org/10.1007/BF02650179

    work page doi:10.1007/bf02650179 1982

  56. [57]

    An invitation to optimal transport, Wasserstein distances, and gradient flows

    Alessio Figalli and Federico Glaudo. An invitation to optimal transport, Wasserstein distances, and gradient flows. 2023

    work page 2023

  57. [58]

    Disgruntled high school athletic director uses AI to clone principal’s voice in racist, antisemitic deep fake

    Ben Finley. “Disgruntled high school athletic director uses AI to clone principal’s voice in racist, antisemitic deep fake”. In: Fortune (Apr. 2024)

    work page 2024

  58. [59]

    Physical Review Letters 85(10), 2200–2203 (2000)

    Shmuel Friedland et al. “Quantum Monge-Kantorovich Problem and Transport Distance between Density Matrices”. In: Physical Review Letters 129.11 (Sept. 2022). issn: 1079-7114. doi: 10.1103/physrevlett. 129.110402. url: http://dx.doi.org/10.1103/PhysRevLett.129.110402

    work page doi:10.1103/physrevlett 2022

  59. [60]

    The Repressive Power of Artificial Intelligence

    Allie Funk, Adrian Shahbaz, and Kian Vesteinsson. The Repressive Power of Artificial Intelligence . Freedom House, 2023

    work page 2023

  60. [61]

    In: Proc

    Bernd G¨ artner and Jiˇ r´ ı Matouˇ sek. “Semidefinite Programming”. In:Approximation Algorithms and Semidef- inite Programming. Springer Berlin Heidelberg, 2012, pp. 15–25.isbn: 978-3-642-22015-9. doi: 10.1007/978- 3-642-22015-9_2 . url: https://doi.org/10.1007/978-3-642-22015-9_2

    work page doi:10.1007/978- 2012

  61. [62]

    Masset, R

    Gy¨ orgy P´ al Geh´ er et al. “Quantum Wasserstein isometries on the qubit state space”. In:Journal of Mathemat- ical Analysis and Applications 522.2 (2023), p. 126955. issn: 0022-247X. doi: https://doi.org/10.1016/j. jmaa.2022.126955. url: https://www.sciencedirect.com/science/article/pii/S0022247X22009696. 37

    work page doi:10.1016/j 2023

  62. [63]

    An entropic interpolation proof of the HWI inequality

    Ivan Gentil et al. “An entropic interpolation proof of the HWI inequality”. In: Stochastic Processes and their Applications 130.2 (2020), pp. 907–923. issn: 0304-4149. doi: https://doi.org/10.1016/j.spa.2019.04

    work page doi:10.1016/j.spa.2019.04 2020

  63. [64]

    url: https://www.sciencedirect.com/science/article/pii/S0304414918303454

    work page

  64. [65]

    Non-commutative Optimal Transport for semi-definite positive ma- trices

    Augusto Gerolin and Nataliia Monina. “Non-commutative Optimal Transport for semi-definite positive ma- trices”. In: arXiv:2309.04846 (Sept. 2023)

    work page arXiv 2023

  65. [66]

    Quantum and Semiquantum Pseudometrics and Applications

    Franccois Golse and Thierry Paul. “Quantum and Semiquantum Pseudometrics and Applications”. In: 2022

    work page 2022

  66. [67]

    Quantum optimal transport is cheaper

    Fran¸ cois Golse, Emanuele Caglioti, and Thierry Paul. “Quantum optimal transport is cheaper”. In: Journal of Statistical Physics 181 (May 2020), pp. 149–162

    work page 2020

  67. [68]

    Onthe mean field and classical limits of quantum mechanics

    Fran¸ cois Golse, Cl´ ement Mouhot, and Thierry Paul. “On the Mean Field and Classical Limits of Quantum Mechanics”. In: Communications in Mathematical Physics 343.1 (Jan. 2016), pp. 165–205. issn: 1432-0916. doi: 10.1007/s00220-015-2485-7 . url: http://dx.doi.org/10.1007/s00220-015-2485-7

    work page doi:10.1007/s00220-015-2485-7 2016

  68. [69]

    The Schr¨ odinger equation in the mean-field and semiclassical regime

    Fran¸ cois Golse and Thierry Paul. “The Schr¨ odinger Equation in the Mean-Field and Semiclassical Regime”. In: Archive for Rational Mechanics and Analysis 223.1 (Aug. 2016), pp. 57–94. issn: 1432-0673. doi: 10. 1007/s00205-016-1031-x . url: http://dx.doi.org/10.1007/s00205-016-1031-x

    work page doi:10.1007/s00205-016-1031-x 2016

  69. [70]

    W ave packets and the q uadratic Monge-Kantorovich distance in quantum mechanics

    Fran¸ cois Golse and Thierry Paul. “Wave packets and the quadratic Monge–Kantorovich distance in quantum mechanics”. In: Comptes Rendus. Math´ ematique356.2 (Jan. 2018), pp. 177–197. issn: 1778-3569. doi: 10. 1016/j.crma.2017.12.007. url: http://dx.doi.org/10.1016/j.crma.2017.12.007

    work page doi:10.1016/j.crma.2017.12.007 2018

  70. [71]

    [HBP23] Aamal Abbas Hussain, Francesco Belardinelli, and G eorgios Piliouras

    Ian Goodfellow et al. “Generative Adversarial Networks”. In: Advances in Neural Information Processing Systems 3 (June 2014). doi: 10.1145/3422622

    work page doi:10.1145/3422622 2014

  71. [72]

    Encoding a qubit in an oscillator

    Daniel Gottesman, Alexei Kitaev, and John Preskill. “Encoding a qubit in an oscillator”. In: Physical Review A 64.1 (June 2001). issn: 1094-1622. doi: 10.1103/physreva.64.012310 . url: http://dx.doi.org/10. 1103/PhysRevA.64.012310

    work page doi:10.1103/physreva.64.012310 2001

  72. [73]

    Concentration inequalities and geometry of convex bodies

    Olivier Gu´ edon, Piotr Nayar, and Tomasz Tkocz. Concentration inequalities and geometry of convex bodies . Jan. 2014

    work page 2014

  73. [74]

    Comparison of parameter estimation methods in stochastic chemical kinetic models: Examples in systems biology

    Ankur Gupta and James B. Rawlings. “Comparison of parameter estimation methods in stochastic chemical kinetic models: Examples in systems biology”. In: AIChE Journal 60.4 (Apr. 2014), pp. 1253–1268. doi: 10.1002/aic.14409

    work page doi:10.1002/aic.14409 2014

  74. [75]

    Quantum marginal problem and incompatibility

    Erkka Haapasalo et al. “Quantum marginal problem and incompatibility”. In: Quantum 5 (June 2021), p. 476. issn: 2521-327X. doi: 10.22331/q-2021-06-15-476 . url: http://dx.doi.org/10.22331/q-2021- 06-15-476

    work page doi:10.22331/q-2021-06-15-476 2021

  75. [76]

    Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware,

    Jonas Haferkamp. “Random quantum circuits are approximate unitary t-designs in depth O nt5+o(1) ”. In: Quantum 6 (Sept. 2022), p. 795. issn: 2521-327X. doi: 10.22331/q- 2022- 09- 08- 795 . url: https: //doi.org/10.22331/q-2022-09-08-795

    work page doi:10.22331/q- 2022

  76. [77]

    Haferkamp, P

    Jonas Haferkamp et al. “Linear growth of quantum circuit complexity”. In: Nature Physics 18.5 (Mar. 2022), pp. 528–532. issn: 1745-2481. doi: 10.1038/s41567-022-01539-6 . url: http://dx.doi.org/10.1038/ s41567-022-01539-6

    work page doi:10.1038/s41567-022-01539-6 2022

  77. [78]

    Quantum optimal transport for approximately finite-dimensional C* algebras

    David F. Hornshaw. “Quantum optimal transport for approximately finite-dimensional C* algebras”. In: arXiv:1910.03312 (2019). url: https://api.semanticscholar.org/CorpusID:203902379

    work page arXiv 1910

  78. [79]

    Quantum coherence and geometric quantum discord

    Ming-Liang Hu et al. “Quantum coherence and geometric quantum discord”. In: Physics Reports 762–764 (Nov. 2018), pp. 1–100. issn: 0370-1573. doi: 10.1016/j.physrep.2018.07.004 . url: http://dx.doi. org/10.1016/j.physrep.2018.07.004

    work page doi:10.1016/j.physrep.2018.07.004 2018

  79. [80]

    Concentration Inequalities for Statistical Inference

    Huiming Zhang Huiming Zhang and Songxi Chen Songxi Chen. “Concentration Inequalities for Statistical Inference”. In: Communications in Mathematical Research 37.1 (Jan. 2021), pp. 1–85. issn: 1674-5647. doi: 10.4208/cmr.2020-0041. url: http://dx.doi.org/10.4208/cmr.2020-0041

    work page doi:10.4208/cmr.2020-0041 2021

  80. [81]

    Enhanced Stability in Quantum Optimal Transport Pseudometrics: From Hartree to Vlasov–Poisson

    Mikaela Iacobelli and Laurent Lafleche. “Enhanced Stability in Quantum Optimal Transport Pseudometrics: From Hartree to Vlasov–Poisson”. In:Journal of Statistical Physics 191.12 (Nov. 2024). issn: 1572-9613. doi: 10.1007/s10955-024-03367-9 . url: http://dx.doi.org/10.1007/s10955-024-03367-9

    work page doi:10.1007/s10955-024-03367-9 2024

Showing first 80 references.