  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.
