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A Dual Formula for the Noncommutative Transport Distance.

Melchior Wirth1

  • 1Institute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria.

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PubMed
Summary

This study explores the noncommutative transport distance and its entropic regularization. We establish a quantum duality formula, extending the Benamou-Brenier formulation for Wasserstein distance using Hamilton-Jacobi-Bellmann equations.

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Area of Science:

  • Mathematical Physics
  • Optimal Transport Theory
  • Noncommutative Geometry

Background:

  • The Carlen-Maas noncommutative transport distance provides a framework for measuring distances in noncommutative spaces.
  • The Becker-Li entropic regularization offers a smoothed version of this distance, crucial for computational and analytical tractability.
  • Wasserstein distances are fundamental in optimal transport, with dual formulations offering powerful analytical tools.

Purpose of the Study:

  • To investigate the properties of the noncommutative transport distance and its entropic regularization.
  • To establish a novel duality formula for the entropic regularization of the noncommutative transport distance.
  • To connect this new formula to existing concepts in optimal transport and Hamilton-Jacobi-Bellmann equations.

Main Methods:

  • Utilizing techniques from optimal transport theory.
  • Applying concepts from functional analysis and partial differential equations.
  • Developing a novel duality approach based on subsolutions of Hamilton-Jacobi-Bellmann equations.

Main Results:

  • A new duality formula for the entropic regularization of the noncommutative transport distance is derived.
  • This formula is shown to be a quantum analogue of the dual Benamou-Brenier formulation of the Wasserstein distance.
  • The formula is expressed in terms of subsolutions to a relevant Hamilton-Jacobi-Bellmann equation.

Conclusions:

  • The derived duality formula offers a new perspective on noncommutative optimal transport.
  • This work bridges concepts from noncommutative geometry, optimal transport, and viscosity solutions.
  • The findings have potential implications for quantum information theory and mathematical physics.