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A quantum version of the Monge-Kantorovich optimal transport problem aims at optimizing a Hermitian cost matrix $C$ over the set of all bipartite coupling states $\rho_{AB}$, s.t. both of its reduced density matrices $\rho_A$ and $\rho_B$ are fixed. I'm interested in its "applications". The classical optimal transport theory can be either used as theoretical tools to study important problems arising in PDE (Schrödinger equation/e Boltzmann equation), metric geometry, random matrices, etc., or used to study problems in practice, e.g. urban planning, economics, data science, etc. So I wish to know whether the quantum optimal transport has similar relevance, from both theoretical and practical perspective.

Any answer, references (survey summarizing its development) and comments are highly appreciated.

PS : Thanks for GJC's comment. I've found some related articles :

https://arxiv.org/pdf/2105.06922.pdf

https://arxiv.org/pdf/1908.01829.pdf

while I still look for such a summary of this field.

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This paper by De Palma et. al. https://ieeexplore.ieee.org/abstract/document/9420734 provides a series of possible applications in the Future perspective section:

  • Quantum State Estimation
  • Robustness of Quantum Machine Learning
  • Quantum Generative Adversarial Networks
  • Quantum Rate Distortion Theory
  • Quantum Differential Privacy
  • Mixing time of Quantum Markov Semigroups
  • Shallow Quantum Circuits
  • Quantum Many-Body Hamiltonians

Also in their other paper https://link.springer.com/article/10.1007/s00023-021-01042-3 implicitly mentioned some of the applications referring to its citations [4-9]:

  • "the study of partial differential equations, by interpreting many evolution equations as gradient flows with respect to transport-induced metrics [4];
  • geometric analysis, with quantitative isoperimetric inequalities [5] and synthetic notions of Ricci curvature bounds [6,7];
  • stochastic analysis in infinite dimensions [8];
  • random combinatorial optimization problems [9];
  • statistics and machine learning [10]."

Although in this second paper none of them are specific applications of the quantum optimal transport but their claim is that it will be a useful tool to address the quantum version of the above problems.

There is also the paper https://arxiv.org/abs/1906.09817 by Ikeda which provides applications for its quantum optimal transport problem by transforming some of these quantum problems into quantum optimal transport:

  • Costly Quantum Walk
  • Quantum Cellular Auto
  • Automata and Games
  • Repeated Quantum Games

I just listed the above items as mentioned in papers. Please look into it further yourself for proper evaluation. Hope it helps.

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