  We consider a channel with a binary input X being corrupted by a
continuous-valued noise that results in a continuous-valued output Y. An
optimal binary quantizer is used to quantize the continuous-valued output Y to
the final binary output Z to maximize the mutual information I(X; Z). We show
that when the ratio of the channel conditional density r(y) = P(Y=y|X=0)/ P(Y
=y|X=1) is a strictly increasing/decreasing function of y, then a quantizer
having a single threshold can maximize mutual information. Furthermore, we show
that an optimal quantizer (possibly with multiple thresholds) is the one with
the thresholding vector whose elements are all the solutions of r(y) = r* for
some constant r* > 0. Interestingly, the optimal constant r* is unique. This
uniqueness property allows for fast algorithmic implementation such as a
bisection algorithm to find the optimal quantizer. Our results also confirm
some previous results using alternative elementary proofs. We show some
numerical examples of applying our results to channels with additive Gaussian
noises.
