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  • Open Access

Critical scaling of the quantum Wasserstein distance

Gonzalo Camacho1,* and Benedikt Fauseweh1,2,†

  • *Contact author: gonzalo.camacho@dlr.de
  • †Contact author: benedikt.fauseweh@tu-dortmund.de
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Phys. Rev. Research 7, 043223 – Published 26 November, 2025

DOI: https://doi.org/10.1103/hbgk-4khg

Abstract

Distinguishing quantum states with minimal sampling overhead is of fundamental importance to teach quantum data to an algorithm. Recently, the quantum Wasserstein distance emerged from the theory of quantum optimal transport as a promising tool in this context. Here, we show on general grounds that the quantum Wasserstein distance between two ground states of a quantum critical system exhibits critical scaling. We demonstrate this explicitly using known closed analytical expressions for the magnetic correlations in the transverse field Ising model, to numerically extract the critical exponents for the distance close to the quantum critical point, confirming our analytical derivation. Our results have implications for learning of ground states of quantum critical phases of matter.

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Phys. Rev. Research 7, 043223– Published 26 November, 2025
  • Received 4 April 2025
  • Revised 15 August 2025
  • Accepted 6 November 2025
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