Entropy Under Additive Bernoulli and Spherical Noises

Let <tex>$Z^{n}$</tex> be iid Bernoulli <tex>$(\delta)$</tex> and <tex>$U^{n}$</tex> be uniform on the set of all binary vectors of weight <tex>$\delta n$</tex> (Hamming sphere). As is well known, the entropies of <tex>$Z^{n}$</tex> and <tex>$U^{n}$</tex> are within <tex>$O(\log n)$</tex>. However, if <tex>$X^{n}$</tex> is another binary random variable independent of <tex>$Z^{n}$</tex> and <tex>$U^{n}$</tex>, we show that <tex>$H(X^{n}+U^{n})$</tex> and <tex>$H(X^{n}+Z^{n})$</tex> are within <tex>$O(\sqrt{n})$</tex> and this estimate is tight. The bound is shown via coupling method. Tightness follows from the observation that the channels <tex>$x^{n}\mapsto x^{n}+U^{n}$</tex> and <tex>$x^{n}\mapsto x^{n}+Z^{n}$</tex> have similar capacities, but the former has zero dispersion. Finally, we show that despite the <tex>$\sqrt{n}$</tex> slack in general, the Mrs. Gerber Lemma for <tex>$H(X^{n}+U^{n})$</tex> holds with only an <tex>$O(\log n)$</tex> correction compared to its brethren for <tex>$H(X^{n}+Z^{n})$</tex>.