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We consider the evolution of variance of entropy (varentropy) in the courseof 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 sideinformation random variable $Y$. The varentropy of $(X,Y)$ is defined as thevariance of the random variable $-\log p_{X|Y}(X|Y)$. A polar transform oforder two is a certain mapping that takes two independent BDEs and produces twonew BDEs that are correlated with each other. It is shown that the sum of thevarentropies at the output of the polar transform is less than or equal to thesum of the varentropies at the input, with equality if and only if at least oneof the inputs has zero varentropy. This result is extended to polar transformsof higher orders and it is shown that the varentropy decreases to zeroasymptotically when the BDEs at the input are independent and identiallydistributed.

Varentropy Decreases Under the Polar Transform