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Wasserstein Geometry of Quantum States and Optimal Transport of Matrix-Valued Measures

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Emerging Applications of Control and Systems Theory

Part of the book series: Lecture Notes in Control and Information Sciences - Proceedings ((LNCOINSPRO))

Abstract

We overview recent results on generalizations of the Wasserstein 2-metric, originally defined on the space of scalar probability densities, to the space of Hermitian matrices and of matrix-valued distributions, as well as some extensions of the theory to vector-valued distributions and discrete spaces (weighted graphs).

This project was supported by ARO grant (W911NF-17-1-0429), AFOSR grants (FA9550-15-1-0045 and FA9550-17-1-0435), NSF (ECCS-1509387), NIH (P41-RR-013218, P41-EB-015902, 1U24CA18092401A1), and a postdoctoral fellowship at MSKCC.

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Notes

  1. 1.

    The domain of \(\nabla _L\) is \({\mathscr {H}}\), hence the identity requires \(XY+YX\), instead of simply XY.

  2. 2.

    The Lindblad term is in the so-called symmetric form since the coefficients are Hermitian.

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Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Iowa State University, Ames, IA, 50011, USA

    Yongxin Chen

  2. Department of Mechanical Engineering, University of California, Irvine, CA, USA

    Tryphon T. Georgiou

  3. Departments Computer Science and Applied Mathematics and Statistics, Stony Brook University, New York City, USA

    Allen Tannenbaum

Authors
  1. Yongxin Chen
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  2. Tryphon T. Georgiou
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  3. Allen Tannenbaum
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Corresponding author

Correspondence to Tryphon T. Georgiou .

Editor information

Editors and Affiliations

  1. IEIIT—CNR, Politecnico di Torino, Turin, Italy

    Roberto Tempo

  2. Department of Systems Engineering, University of Texas at Dallas, Richardson, Texas, USA

    Stephen Yurkovich

  3. Russ Engineering Center, Wright State University, Dayton, Ohio, USA

    Pradeep Misra

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Chen, Y., Georgiou, T.T., Tannenbaum, A. (2018). Wasserstein Geometry of Quantum States and Optimal Transport of Matrix-Valued Measures. In: Tempo, R., Yurkovich, S., Misra, P. (eds) Emerging Applications of Control and Systems Theory. Lecture Notes in Control and Information Sciences - Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-319-67068-3_10

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  • DOI: https://doi.org/10.1007/978-3-319-67068-3_10

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  • Print ISBN: 978-3-319-67067-6

  • Online ISBN: 978-3-319-67068-3

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