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Quantum Optimal Transport is Cheaper

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Abstract

We compare bipartite (Euclidean) matching problems in classical and quantum mechanics. The quantum case is treated in terms of a quantum version of the Wasserstein distance. We show that the optimal quantum cost can be cheaper than the classical one. We treat in detail the case of two particles: the equal mass case leads to equal quantum and classical costs. Moreover, we show examples with different masses for which the quantum cost is strictly cheaper than the classical cost.

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Notes

  1. Note the unessential difference with the definition of the cost C in [7,8,9] created by the shift \(-2d\hbar \) and accounts for a shift by \(2d\hbar \) in the two next formulas.

  2. Here also, we use a different normalization than the one in [7,8,9], since we deal exclusively with density matrices. With the present normalization, one has \({\text {trace}}{T}=\int _{\mathbf {R}^{2d}}\tau (dq,dp)\).

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Acknowledgements

This work has been partially carried out thanks to the support of the LIA AMU-CNRS-ECM-INdAM Laboratoire Ypatie des Sciences Mathématiques (LYSM).

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Correspondence to E. Caglioti.

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Communicated by Eric A. Carlen.

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Caglioti, E., Golse, F. & Paul, T. Quantum Optimal Transport is Cheaper. J Stat Phys 181, 149–162 (2020). https://doi.org/10.1007/s10955-020-02571-7

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