Journal
IEEE TRANSACTIONS ON AUTOMATIC CONTROL
Volume 63, Issue 8, Pages 2612-2619Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TAC.2017.2767707
Keywords
Matrix optimal transport; quantum mechanics; entropic flows; non-commutative Wasserstein
Funding
- Air Force Office of Scientific Research [FA9550-15-1-0045, FA9550-17-1-0435]
- ARO [W911NF-17-1-0429]
- National Center for Research Resources [P41-RR-013218]
- National Institute of Biomedical Imaging and Bioengineering [P41-EB-015902]
- National Science Foundation
- National Institutes of Health [P30-CA-008748]
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In this paper, we describe a possible generalization of the Wasserstein-2 metric, originally defined on the space of scalar probability densities, to the space of Hermitian matrices with trace one and to the space of matrix-valued probability densities. Our approach follows a control-theoretic optimization formulation of the Wasserstein-2 metric, having its roots in fluid dynamics, and utilizes certain results from the quantum mechanics of open systems, in particular the Lindblad equation. It allows determining the gradient flow for the quantum entropy relative to this matricial Wasserstein metric.
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