  We consider the problem of estimating the mean of a normal distribution under
the following constraint: the estimator can access only a single bit from each
sample from this distribution. We study the squared error risk in this
estimation as a function of the number of samples and one-bit measurements $n$.
We consider an adaptive estimation setting where the single-bit sent at step
$n$ is a function of both the new sample and the previous $n-1$ acquired bits.
For this setting, we show that no estimator can attain asymptotic mean squared
error smaller than $\pi/(2n)+O(n^{-2})$ times the variance. In other words,
one-bit restriction increases the number of samples required for a prescribed
accuracy of estimation by a factor of at least $\pi/2$ compared to the
unrestricted case. In addition, we provide an explicit estimator that attains
this asymptotic error, showing that, rather surprisingly, only $\pi/2$ times
more samples are required in order to attain estimation performance equivalent
to the unrestricted case.
