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 $\widehat{x}$ and $\widehat{p}$, the quantum entropic uncertainty relation can take a suggestive form, where the standard deviations ${\sigma }_x$ and ${\sigma }_p$ 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, ${\sigma }_x{\sigma }_p\ge \hslash e^{\mathcal{N}}/2$; (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.

• Received 20 April 2015

DOI:https://doi.org/10.1103/PhysRevA.92.012114