  We consider the evolution of variance of entropy (varentropy) in the course
of a polar transform operation on binary data elements (BDEs). A BDE is a pair
$(X,Y)$ consisting of a binary random variable $X$ and an arbitrary side
information random variable $Y$. The varentropy of $(X,Y)$ is defined as the
variance of the random variable $-\log p_{X|Y}(X|Y)$. A polar transform of
order two is a certain mapping that takes two independent BDEs and produces two
new BDEs that are correlated with each other. It is shown that the sum of the
varentropies at the output of the polar transform is less than or equal to the
sum of the varentropies at the input, with equality if and only if at least one
of the inputs has zero varentropy. This result is extended to polar transforms
of higher orders and it is shown that the varentropy decreases to zero
asymptotically when the BDEs at the input are independent and identially
distributed.
