A refinement of the Robertson-Schr\"odinger uncertainty principle and a Hirschman-Shannon inequality for Wigner distributions

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.