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Quantum Monge-Kantorovich Problem and Transport Distance between Density Matrices.

Shmuel Friedland1, Michał Eckstein2, Sam Cole3

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This summary is machine-generated.

This study analyzes quantum optimal transport, defining a new quantum semidistance between quantum states. This distance is bounded by known measures and offers applications in quantum machine learning.

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Area of Science:

  • Quantum Information Theory
  • Quantum Computing
  • Mathematical Physics

Background:

  • The Monge-Kantorovich optimal transport problem is a fundamental concept in mathematics and economics.
  • Quantum information theory seeks to understand and utilize quantum mechanical phenomena for information processing.

Purpose of the Study:

  • To analyze a quantum version of the Monge-Kantorovich optimal transport problem.
  • To define and investigate a novel quantum semidistance measure between quantum states.

Main Methods:

  • Minimizing transport cost over bipartite coupling states with fixed reduced density matrices.
  • Utilizing a quantum cost matrix proportional to the projector on the antisymmetric subspace.
  • Deriving semianalytic expressions for optimal transport cost in the single-qubit case.

Main Results:

  • The minimal transport cost yields a semidistance bounded by Bures distance and root infidelity.
  • The square root of the optimal transport cost in the single-qubit case forms a Wasserstein distance analog.
  • Introduced 'swap fidelity' as a measure of quantum state proximity.

Conclusions:

  • The developed quantum optimal transport framework provides a new tool for quantifying distances between quantum states.
  • The introduced swap fidelity has potential applications in quantum machine learning and quantum information processing.