  In this letter we study the weak-convergence properties of random variables
generated by unsharp quantum measurements. More precisely, for a sequence of
random variables generated by repeated unsharp quantum measurements, we study
the limit distribution of relative frequency. We provide the de Finetti-type
representation theorem for all separable states, showing that the distribution
can be well approximated by mixture of normal distributions. No symmetry
restrictions, such as the permutational invariance were needed. Furthermore, we
investigate the convergence rates and show that the relative frequency can
stabilize to some constant at best at the rate of order $1/\sqrt{N}$ for all
separable inputs. On the other hand, we provide an example of a strictly
unsharp quantum measurement where the better rates are achieved by using
entangled inputs. This means that in certain cases the noise generated by the
measurement process can be suppressed by using entanglement. We deliver our
result in the form of quantum information task where the player achieves the
goal with certainty in the limiting case by using entangled inputs or fails
with certainty by using separable inputs.
