Suppose $( X_{n} )$ is an ergodic binary process with $$ \mathbb P(X_{n}=1)= \mathbb P(X_{n}=0)=\frac 12. $$

Naturally the entropy (rate) $h(X)$ of $X=(X_{n})$ satisfies $$ h(X)=\lim_{n\to\infty} \frac 1n H(X_1,\ldots, X_n)\le H(\frac 12)=\log 2. $$ The entropy will "drop" due to dependencies in $(X_n)$.

Suppose that $$ \sigma^2=\sum_{n=2}^{\infty} \sup_{a,b\in\{0,1\} }\Bigl| \mathbb P(X_{1}=a,X_{n}=b)-\mathbb P(X_{1}=a)\mathbb P(X_{n}=b)\Bigr| $$ is very small: $\sigma^2\ll 1$. Small $\sigma^2$ suggests that $(X_{n})$ are nearly independent. Can one derive a lower bound for $h(X)$ of the form $$ h(X)\ge \log 2- f(\sigma^2), $$ where $f$ is a continuous function with $f(0)=0$? My feeling is that small $\sigma^2$ should imply that the $\bar{d}$-metric between $\{X_{n}\}$ and the $\text{Ber}(1/2)$-process should be small, hence implying that entropies should be close as well.

In the oposite direction, suppose $\sigma^2$ is large. Then the entropy entropy drop $\log 2- h(X)$ should be substantial as well. For example, for every $\epsilon>0$ one can find $D>0$ such that for every binary process with $\sigma^2>D$ $$ h(X)\le \log 2 -\epsilon. $$

Any ideas, references?