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

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, ${\sigma }_x{\sigma }_p\ge \hslash e^{\mathcal{N}}/2$; (ii) the additional factor $\mathcal{N}$ 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.

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References in corpus (4)

• Heisenberg's Uncertainty Principle
• Quantifying the non-Gaussian character of a quantum state by quantum relative entropy
• Quantum and classical entropic uncertainty relations
• Optimized entropic uncertainty relation for successive measurement

Cited by in corpus (6)

• Continuous-variable entropic uncertainty relations
• Entropy-power uncertainty relations : towards a tight inequality for all Gaussian pure states
• Relative entropic uncertainty relation
• Non-Gaussianity and entropy-bounded uncertainty relations: Application to detection of non-Gaussian entangled states
• Relative entropic uncertainty relation for scalar quantum fields
• Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements