  Recent advances have linked various statements involving sumsets and
cardinalities with corresponding statements involving sums of random variables
and entropies. In this vein, this paper shows that the quantity $2{\bf H}\{X,
Y\} - {\bf H}\{X+Y\}$ is a natural entropic analogue of the additive energy
$E(A,B)$ between two sets. We develop some basic theory surrounding this
quantity, and demonstrate its role in the proof of Tao's entropy variant of the
Balog--Szemer\'edi--Gowers theorem. We examine the regime where entropic
additive energy is small, and discuss a family of random variables related to
Sidon sets. In finite fields, one can define an entropic multiplicative energy
as well, and we formulate sum-product-type conjectures relating these two
entropic energies.
