Quantum Polar Duality and the Symplectic Camel: a Geometric Approach to Quantization
arXiv:2009.10678
Abstract
We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke-Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho--Stark principle from the point of view of quantum polarity.
Typos corrected. To appear in Foundations of Physics
References in corpus (6)
- Experimental Quantum State Tomography of Optical Fields and Ultrafast Statistical Sampling
- Unitarily localizable entanglement of Gaussian states
- Quantum States and Hardy's Formulation of the Uncertainty Principle : a Symplectic Approach
- Quantum Harmonic Analysis of the Density Matrix
- When symplectic topology meets Banach space geometry
- Quantum uncertainty relation based on the mean deviation