Role of quantum non-Gaussian distance in entropic uncertainty relations

Abstract

A Gaussian distribution of a quantum state with continuous spectra is known to maximize the Shannon entropy at a fixed variance. Applying it to a pair of canonically conjugate quantum observables $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}$, the quantum entropic uncertainty relation can take a suggestive form, where the standard deviations ${\ensuremath{\sigma}}_{x}$ and ${\ensuremath{\sigma}}_{p}$ are featured explicitly. From the construction of the entropic uncertainty relation, it follows in a transparent manner that (i) the entropic uncertainty relation implies the Kennard-Robertson uncertainty relation in a modified form, ${\ensuremath{\sigma}}_{x}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}\ensuremath{\hbar}{e}^{\mathcal{N}}/2$; (ii) the additional factor $\mathcal{N}$ quantifies the quantum non-Gaussianity of the probability distributions of two observables; and (iii) the lower bound of the entropic uncertainty relation for a non-Gaussian continuous-variable (CV) mixed state becomes stronger with purity. The optimality of specific non-Gaussian CV states for the refined uncertainty relation has been investigated and the existence of a new class of CV quantum state is identified.