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

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. The authors were supported under the EU-project MEXT-CT-2004-51715.