  Let $X$ and $Y$ be dependent random variables. This paper considers the
problem of designing a scalar quantizer for $Y$ to maximize the mutual
information between the quantizer's output and $X$, and develops fundamental
properties and bounds for this form of quantization, which is connected to the
log-loss distortion criterion. The main focus is the regime of low $I(X;Y)$,
where it is shown that, if $X$ is binary, a constant fraction of the mutual
information can always be preserved using $\mathcal{O}(\log(1/I(X;Y)))$
quantization levels, and there exist distributions for which this many
quantization levels are necessary. Furthermore, for larger finite alphabets $2
< |\mathcal{X}| < \infty$, it is established that an $\eta$-fraction of the
mutual information can be preserved using roughly $(\log(| \mathcal{X} |
/I(X;Y)))^{\eta\cdot(|\mathcal{X}| - 1)}$ quantization levels.
