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On the matrix Monge–Kantorovich problem

Part of: Miscellaneous topics in calculus of variations and optimal control General mathematical topics and methods in quantum theory Topological linear spaces and related structures

Published online by Cambridge University Press:  05 August 2019

YONGXIN CHEN ,
WILFRID GANGBO ,
TRYPHON T. GEORGIOU  and
ALLEN TANNENBAUM Open the ORCID record for ALLEN TANNENBAUM [Opens in a new window]
YONGXIN CHEN
Affiliation:
School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, USA email: [email protected]
WILFRID GANGBO
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA, USA email: [email protected]
TRYPHON T. GEORGIOU
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA email: [email protected]
ALLEN TANNENBAUM
Affiliation:
Departments of Computer Science and Applied Mathematics & Statistics, Stony Brook University, NY, USA email: [email protected]
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Abstract

The classical Monge–Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the Wasserstein distance. In this note, we formulate a natural matrix counterpart of the MK problem for positive-definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimisation problem, strong duality, an analogue of the Poincaré–Wirtinger inequality and a Lax–Hopf–Oleinik–type result.

Keywords

Optimal mass transportquantum mechanicsWasserstein distance

MSC classification

Primary: 46A20: Duality theory
Secondary: 49N15: Duality theory 81Q35: Quantum mechanics on special spaces
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 4 , August 2020 , pp. 574 - 600
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Copyright
© Cambridge University Press 2019

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Footnotes

†

This project was supported by AFOSR grants (FA9550-17-1-0435 and FA9550-18-1-0502), grants from the National Center for Research Resources (P41-RR-013218) and the National Institute of Biomedical Imaging and Bioengineering (P41-EB-015902), National Science Foundation grants (DMS-1160939, 1665031, 1807664, 1839441 and 1901599), grants from the National Institutes of Health (1U24CA18092401A1 and R01-AG048769) and a postdoctoral fellowship through Memorial Sloan Kettering Cancer Center.

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