  Conventionally, unknown quantum states are characterized using quantum-state
tomography based on strong or weak measurements carried out on an ensemble of
identically prepared systems. By contrast, the use of protective measurements
offers the possibility of determining quantum states from a series of weak,
long measurements performed on a single system. Because the fidelity of a
protectively measured quantum state is determined by the amount of state
disturbance incurred during each protective measurement, it is crucial that the
initial quantum state of the system is disturbed as little as possible. Here we
show how to systematically minimize the state disturbance in the course of a
protective measurement, thus enabling the maximization of the fidelity of the
quantum-state measurement. Our approach is based on a careful tuning of the
time dependence of the measurement interaction and is shown to be dramatically
more effective in reducing the state disturbance than the previously considered
strategy of weakening the measurement strength and increasing the measurement
time. We describe a method for designing the measurement interaction such that
the state disturbance exhibits polynomial decay to arbitrary order in the
inverse measurement time $1/T$. We also show how one can achieve even faster,
subexponential decay, and we find that it represents the smallest possible
state disturbance in a protective measurement. In this way, our results show
how to optimally measure the state of a single quantum system using protective
measurements.
