Letters in Mathematical Physics

, Volume 80, Issue 1, pp 69-82

First online:

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

  • Maurice de GossonAffiliated withFaculty of Mathematics, University of Vienna Email author 
  • , Franz LuefAffiliated withFaculty of Mathematics, University of Vienna

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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