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

Abstract

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.

To appear in Lett. Math. Phys

Cited by in corpus (13)

• Wigner Measures in Noncommutative Quantum Mechanics
• Quantum Harmonic Analysis of the Density Matrix
• Mixed Quantum States with Variable Planck Constant
• The volume of Gaussian states by information geometry
• On the Usefulness of Modulation Spaces in Deformation Quantization
• On the Use of Minimum Volume Ellipsoids and Symplectic Capacities for Studying Classical Uncertainties for Joint Position-Momentum Measurements
• The Multi-Dimensional Hardy Uncertainty Principle and its Interpretation in Terms of the Wigner Distribution; Relation With the Notion of Symplectic Capacity
• A new approach to the star-genvalue equation
• The uncertainty principle and the energy identity for holomorphic maps in geometric quantum mechanics
• Quantum Polar Duality and the Symplectic Camel: a Geometric Approach to Quantization
• Mean-dispersion principles and the Wigner transform
• Polar Duality Between Pairs of Transverse Lagrangian Planes. Applications to Uncertainty Principles
• On the Dependence of Quantum States on the Value of Planck's Constant