Letters in Mathematical Physics

, Volume 80, Issue 1, pp 69–82

Quantum States and Hardy’s Formulation of the Uncertainty Principle: a Symplectic Approach


DOI: 10.1007/s11005-007-0150-6

Cite this article as:
de Gosson, M. & Luef, F. Lett Math Phys (2007) 80: 69. doi:10.1007/s11005-007-0150-6


We express the condition for a phase space Gaussian to be the Wigner distribution of a mixed quantum state in terms of the symplectic capacity of the associated Wigner ellipsoid. Our results are motivated by Hardy’s formulation of the uncertainty principle for a function and its Fourier transform. As a consequence we are able to state a more general form of Hardy’s theorem.

Mathematics Subject Classification (2000)

81S10 81S30 37J05 


Hardy’s uncertainty principle Wigner distribution density operator positivity 


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Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria

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