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Quantum State Assignment Flows

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Scale Space and Variational Methods in Computer Vision (SSVM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14009))

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Abstract

This paper extends the assignment flow approach from categorial distributions to complex-valued Hermitian density matrices, used as state spaces for representing and analyzing data associated with vertices of an underlying graph. Determining the flow of the resulting dynamical system by geometric integration causes an interaction of these non-commuting states across the graph, and the assignment of a pure (rank-one) state to each vertex after convergence. Experiments with toy systems indicate the potential of the novel approach for data representation and analysis.

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Notes

  1. 1.

    We keep the symbol \(\pi _{0}\) for notational simplicity. The argument disambiguates the projections (2.3) and (2.11), respectively.

  2. 2.

    The use of the symbol S in the present context should not be confused with the similarity mapping (3.13). We just adhere to the notation used in prior work in order to reference clearly.

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Acknowledgements

This work is funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster). This work is funded by the Deutsche Forschungsgemeinschaft (DFG), grant SCHN 457/17-1, within the priority programme SPP 2298: “Theoretical Foundations of Deep Learning”.

Author information

Authors and Affiliations

  1. Kichhoff Institute for Physics, Heidelberg University, Heidelberg, Germany

    Martin Gärttner

  2. Mathematical Institute, Heidelberg University, Heidelberg, Germany

    Peter Albers

  3. Institute for Applied Mathematics, Heidelberg University, Heidelberg, Germany

    Jonathan Schwarz, Bastian Boll, Daniel Gonzalez-Alvarado, Dmitrij Sitenko & Christoph Schnörr

Authors
  1. Jonathan Schwarz
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  2. Bastian Boll
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  3. Daniel Gonzalez-Alvarado
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  5. Martin Gärttner
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  6. Peter Albers
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  7. Christoph Schnörr
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Corresponding author

Correspondence to Jonathan Schwarz .

Editor information

Editors and Affiliations

  1. CNRS, Université Côte d'Azur, Sophia-Antipolis, France

    Luca Calatroni

  2. University of Insubria, Como, Italy

    Marco Donatelli

  3. University of Bologna, Bologna, Italy

    Serena Morigi

  4. University of Modena and Reggio Emilia, Modena, Italy

    Marco Prato

  5. University of Genova, Genova, Italy

    Matteo Santacesaria

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Schwarz, J. et al. (2023). Quantum State Assignment Flows. In: Calatroni, L., Donatelli, M., Morigi, S., Prato, M., Santacesaria, M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2023. Lecture Notes in Computer Science, vol 14009. Springer, Cham. https://doi.org/10.1007/978-3-031-31975-4_57

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  • DOI: https://doi.org/10.1007/978-3-031-31975-4_57

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-31974-7

  • Online ISBN: 978-3-031-31975-4

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