  When estimating an unknown single pure qubit state, the optimum fidelity is
2/3. As it is well known, the value 2/3 can be achieved in one step, by a
single ideal measurement of the polarization along a random direction. I
analyze the opposite strategy which is the long sequence of unsharp
polarization measurements. The evolution of the qubit under the influence of
repeated measurements is quite complicated in the general case. Fortunately, in
a certain limit of very unsharp measurements the qubit will obey simple
stochastic evolution equations known for long under the name of time-continuous
measurement theory. I discuss how the outcomes of the very unsharp measurements
will asymptotically contribute to our knowledge of the original qubit. It is
reassuring that the fidelity will achieve the optimum 2/3 for long enough
sequences of the unsharp measurements.
