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

TL;DR

This paper introduces a stronger refinement of the Robertson-Schrodinger uncertainty principle using Wigner distributions, which is saturated only by Gaussian states and connects to harmonic analysis, entropy, and quantum state purity.

Contribution

The paper presents a new refined RSUP based on Wigner distributions, stronger than the standard version, and establishes related entropy and inequality results for quantum states.

Findings

Refined RSUP is saturated only by pure Gaussian Wigner functions.

Derived inequalities involving entropy and covariance matrix of Wigner distributions.

Sharp estimates depend on the purity of the quantum state and are saturated by Gaussian states.

Abstract

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.