Consider the following protocol:

Alice has a number $X$, chosen according to a known distribution $\mathcal D$ (e.g., $X\sim U[0,1]$).

She can send a bit to Bob, giving him more information about $X$ (e.g., she can send $Y=\mbox{Bernoulli}(X)$. In turn, Bob estimates $X$ (e.g., as $\widehat X = Y$).

I'm interested in lower bounding the expected (over the choice of $X$) variance of any such protocol.

In the above example, we have $\mbox{Var}[\widehat X | X] = X(1-X)$ and thus $$\mathbb E[\mbox{Var}[\widehat X]] = \mathbb E[\mbox{Var}[\widehat x | X]] = \int_0^1x(1-x)dx = 1/6.$$

For this specific distribution (uniform), this protocol seems to yield the lowest possible expected variance for any unbiased estimation.

Can we use information-theoretic arguments to lower bound the variance for different distributions $\mathcal D$ for any protocol?

I looked into applying conditional differential entropy, but this seems to apply to a specific choice of protocol (how to choose which bit to send).