Single qubit estimation from repeated unsharp measurements

Abstract

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.