## A refinement of the Robertson-Schr\"odinger uncertainty principle and a Hirschman-Shannon inequality for Wigner distributions. (arXiv:1712.09475v1 [math-ph])

We propose a refinement of the Robertson-Schrodinger uncertainty principle
(RSUP) using Wigner distributions. This new principle is stronger than the
RSUP. In particular, and unlike the RSUP, which can be saturated by many phase
space functions, the refined RSUP can be saturated by pure Gaussian Wigner
functions only. Moreover, the new principle is technically as simple as the
standard RSUP. In addition, it makes a direct connection with modern harmonic
analysis, since it involves the Wigner transform and its symplectic Fourier
transform, which is the radar ambiguity function. As a by-product of the
refined RSUP, we derive inequalities involving the entropy and the covariance
matrix of Wigner distributions. These inequalities refine the Shanon and the
Hirschman inequalities for the Wigner distribution of a mixed quantum state
$\rho$. We prove sharp estimates which critically depend on the purity of
$\rho$ and which are saturated in the Gaussian case.查看全文