Qubit purification speed-up for three complementary continuous measurements

We consider qubit purification under simultaneous continuous measurement of the three non-commuting qubit operators \sigma_x, \sigma_y, \sigma_z. The purification dynamics is quantified by (i) the average purification rate, and (ii) the mean time of reaching given level of purity, (1-\epsilon). Under ideal measurements (detector efficiency \eta=1), we show in the first case an asymptotic mean purification speed-up of 4 as compared to a standard (classical) single-detector measurement. However by the second measure --- the mean time of first passage T(\epsilon) of the purity --- the corresponding speed-up is only 2. We explain these speed-ups using the isotropy of the qubit evolution that provides an equivalence between the original measurement directions and three simultaneous measurements, one with an axis aligned along the Bloch vector and the other with axes in the two complementary directions. For inefficient detectors, \eta=1-\delta <1 the mean time of first passage T(\delta,\epsilon) increases since qubit purification competes with an isotropic qubit dephasing. In the asymptotic high-purity limit (\epsilon, \delta << 1) we show that the increase possesses a scaling behavior: \Delta T(\delta,\epsilon) is a function only of the ratio {\delta}/{\epsilon}. The increase \Delta T({\delta}/{\epsilon}) is linear for small argument but becomes exponential ~ exp({\delta}/2{\epsilon}) for {\delta}/{\epsilon} large.