The role of quantum non-Gaussian distance in entropic uncertainty relation

Gaussian distribution of a quantum state with continuous spectrum is known to maximize the Shannon entropy at a fixed variance. Applying it to a pair of canonically conjugate quantum observables $hat x$ and $hat p$, quantum entropic uncertainty relation can take a suggestive form, where the standard deviations $sigma_x$ and $sigma_p$ are featured explicitly. From the construction, it follows in a transparent manner that: (i) the entropic uncertainty relation implies the Kennard-Robertson uncertainty relation in a modifed form, $sigma_xsigma_pgeqhbar e^{cal N}/2$; (ii) the additional factor ${cal N}$ quantifies the quantum non-Gaussianity of the probability distributions of two observables; (iii) the lower bound of the entropic uncertainty relation for non-gaussian continuous variable (CV) mixed state becomes stronger with purity. Optimality of specific non-gaussian CV states to the refined uncertainty relation has been investigated and the existance of new class of CV quantum state is identified.