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Geometry of the Set of Mixed Quantum States: An Apophatic Approach

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Geometric Methods in Physics

Part of the book series: Trends in Mathematics ((TM))

Abstract

The set of quantum states consists of density matrices of order N, which are hermitian, positive and normalized by the trace condition. We analyze the structure of this set in the framework of the Euclidean geometry naturally arisingin the space of hermitian matrices. For \( N\,\,=\,\,2 \) this set is the Bloch ball, embedded in \( \mathbb{R}^3 \). For \( N\,\,\geq \,\,3\) this set of dimensionality \( N^2 \,\,-\,1 \) has a much richer structure. We study its properties and at first advocate an apophatic approach, which concentrates on characteristics not possessed by this set. We also apply more constructive techniques and analyze twodimensional cross-sections and projections of the set of quantum states. They are dual to each other. At the end we make some remarks on certain dimension dependent properties.

Mathematics Subject Classification (2010). Primary 81P16; Secondary 52A20.

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Author information

Authors and Affiliations

  1. Stockholms Universitet, Fysikum Roslagstullsbacken 21, S-106 91, Stockholm, Sweden

    Ingemar Bengtsson

  2. Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, D-04103, Leipzig, Germany

    Stephan Weis

  3. Institute of Physics, Jagiellonian University, ul. Reymonta 4, PL-30-059, Kraków, Poland

    Karol Życzkowski

  4. Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, PL-02-668, Warsaw, Poland

    Karol Życzkowski

Authors
  1. Ingemar Bengtsson
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  2. Stephan Weis
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  3. Karol Życzkowski
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Corresponding author

Correspondence to Ingemar Bengtsson .

Editor information

Editors and Affiliations

  1. Departamento de Física, CINVESTAV, Mexico City, Distrito Federal, Mexico

    Piotr Kielanowski

  2. Dept. of Mathematics and Statistics, Concordia University, Montreal, Québec, Canada

    S. Twareque Ali

  3. Institute of Mathematics, University of Bialystok, Bialystok, Poland

    Anatol Odzijewicz

  4. Mathematics Research Unit, FSTC, University of Luxembourg, Luxembourg-Kirchberg, Luxembourg

    Martin Schlichenmaier

  5. School of Mathematics, University of Manchester, Manchester, United Kingdom

    Theodore Voronov

Additional information

Dedicated to prof. Bogdan Mielnik on the occasion of his 75th birthday

Primary 81P16; Secondary 52A20.

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Bengtsson, I., Weis, S., Życzkowski, K. (2013). Geometry of the Set of Mixed Quantum States: An Apophatic Approach. In: Kielanowski, P., Ali, S., Odzijewicz, A., Schlichenmaier, M., Voronov, T. (eds) Geometric Methods in Physics. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0448-6_15

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