Continuous-variable entropic uncertainty relations

TL;DR

This paper reviews recent developments in entropic uncertainty relations for continuous variables in quantum physics, emphasizing their reformulation via entropy power and introducing new, tighter relations that account for correlations.

Contribution

It provides a comprehensive review of continuous-variable entropic uncertainty relations and introduces a novel, tighter relation for correlated variables, with conjectures for multi-variable cases.

Findings

Reformulation of uncertainty relations using entropy power.

Introduction of a tighter uncertainty relation for correlated variables.

Conjectures for entropic relations involving more than two variables.

Abstract

Uncertainty relations are central to quantum physics. While they were originally formulated in terms of variances, they have later been successfully expressed with entropies following the advent of Shannon information theory. Here, we review recent results on entropic uncertainty relations involving continuous variables, such as position $x$ and momentum $p$. This includes the generalization to arbitrary (not necessarily canonically-conjugate) variables as well as entropic uncertainty relations that take $x$-$p$ correlations into account and admit all Gaussian pure states as minimum uncertainty states. We emphasize that these continuous-variable uncertainty relations can be conveniently reformulated in terms of entropy power, a central quantity in the information-theoretic description of random signals, which makes a bridge with variance-based uncertainty relations. In this review, we take the quantum optics viewpoint and consider uncertainties on the amplitude and phase quadratures of the electromagnetic field, which are isomorphic to $x$ and $p$, but the formalism applies to all such variables (and linear combinations thereof) regardless of their physical meaning. Then, in the second part of this paper, we move on to new results and introduce a tighter entropic uncertainty relation for two arbitrary vectors of intercommuting continuous variables that take correlations into account. It is proven conditionally on reasonable assumptions. Finally, we present some conjectures for new entropic uncertainty relations involving more than two continuous variables.