期刊
JOURNAL OF FUNCTIONAL ANALYSIS
卷 282, 期 9, 页码 -出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jfa.2022.109417
关键词
Quantum Wasserstein metric; Triangle inequality; Kantorovich duality; Optimal transport
类别
This paper proves variants of the triangle inequality for the quantum analogues of the Wasserstein metric, introduces a different argument compared to the classical proof utilizing a Kantorovich duality analogy, and defines an analogue of the Brenier transport map.
This paper proves variants of the triangle inequality for the quantum analogues of the Wasserstein metric of exponent 2 introduced in Golse et al. (2016) [13] to compare two density operators, and in Golse and Paul (2017) [14] to compare a phase space probability measure and a density operator. The argument differs noticeably from the classical proof of the triangle inequality for Wasserstein metrics, which is based on a disintegration theorem for probability measures, and uses in particular an analogue of the Kantorovich duality for the functional defined in Golse and Paul (2017) [14]. Finally, this duality theorem is used to define an analogue of the Brenier transport map for the functional defined in Golse and Paul (2017) [14] to compare a phase space probability measure and a density operator. (C) 2022 Published by Elsevier Inc.
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