Quantum Polar Duality and the Symplectic Camel: a Geometric Approach to Quantization

TL;DR

This paper introduces quantum polarity, a geometric Fourier transform linking position and momentum sets, revealing deep connections between quantum uncertainty, symplectic geometry, and convex analysis, with implications for quantum state reconstruction.

Contribution

It defines quantum polarity and demonstrates its role in solving the Pauli reconstruction problem for Gaussian states, connecting quantum mechanics with convex and symplectic geometry.

Findings

Quantum polarity links position and momentum sets via a geometric Fourier transform.

It solves the Pauli reconstruction problem for Gaussian wavefunctions.

The work relates quantum uncertainty principles to convex geometric inequalities.

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-Santal\'o 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.