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Geometry and Structure of Quantum Phase Space

Abstract

The application of geometry to physics has provided us with new insightful information about many physical theories such as classical mechanics, general relativity, and quantum geometry (quantum gravity). The geometry also plays an important role in foundations of quantum mechanics and quantum information. In this work we discuss a geometric framework for mixed quantum states represented by density matrices, where the quantum phase space of density matrices is equipped with a symplectic structure, an almost complex structure, and a compatible Riemannian metric. This compatible triple allow us to investigate arbitrary quantum systems. We will also discuss some applications of the geometric framework.

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Acknowledgments

The author acknowledges useful comments and discussions with Ole Andersson at Stockholm University and also discussion with Faisal Shah Khan at Khalifa University. The author also acknowledges the financial support from the Swedish Research Council (VR).

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Correspondence to Hoshang Heydari.

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Heydari, H. Geometry and Structure of Quantum Phase Space. Found Phys 45, 851–857 (2015). https://doi.org/10.1007/s10701-015-9907-4

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  • DOI: https://doi.org/10.1007/s10701-015-9907-4

Keywords

  • Quantum mixed states
  • Density operators
  • Quantum phase space
  • Uncertainty relation
  • Geometry