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Quantum Monge-Kantorovich Problem and Transport Distance between Density Matrices
Shmuel Friedland1, Michał Eckstein2, Sam Cole3
1Department of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, Illinois 60607-7045, USA.
Physical Review Letters
|September 26, 2022
Summary
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.
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.
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