Smoothly leaking information over time
Via Michael Kass: let $Y(t)$ be a Wiener process starting at $Y(0) = X$, and define

$$f(t) = H(X~|~Y(t))$$

Then $f(0) = 0$, $f(\infty)=1$, and $f(t)$ is smoothly strictly increasing in between.  Thus, for any $k$ we can find $0 \le t \le \infty$ s.t. $X_k = Y(t)$ has the desired conditional entropy ($t$ will be a decreasing function of $k$).