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

TL;DR

This paper explores how quantum non-Gaussianity influences entropic uncertainty relations, revealing that non-Gaussian states can tighten uncertainty bounds and identifying new classes of such states.

Contribution

It introduces a quantum non-Gaussian distance as a key factor in entropic uncertainty, linking non-Gaussianity to stronger uncertainty bounds and discovering new non-Gaussian CV states.

Findings

Quantum non-Gaussianity quantifies uncertainty bounds.

Stronger bounds for non-Gaussian mixed states with higher purity.

Existence of new classes of non-Gaussian CV quantum states.

Abstract

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, $\sigma_x\sigma_p\geq\hbar e^{\cal N}/2$; (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.