  We consider the fundamental problem of communicating an estimate of a real
number $x\in[0,1]$ using a single bit. A sender that knows $x$ chooses a value
$X\in\set{0,1}$ to transmit. In turn, a receiver estimates $x$ based on the
value of $X$. We consider both the biased and unbiased estimation problems and
aim to minimize the cost. For the biased case, the cost is the worst-case (over
the choice of $x$) expected squared error, which coincides with the variance if
the algorithm is required to be unbiased.
  We first overview common biased and unbiased estimation approaches and prove
their optimality when no shared randomness is allowed. We then show how a small
amount of shared randomness, which can be as low as a single bit, reduces the
cost in both cases. Specifically, we derive lower bounds on the cost attainable
by any algorithm with unrestricted use of shared randomness and propose
near-optimal solutions that use a small number of shared random bits. Finally,
we discuss open problems and future directions.
