  Defining and measuring the error of a measurement is one of the most
fundamental activities in experimental science. However, quantum theory shows a
peculiar difficulty in extending the classical notion of root-mean-square (rms)
error to quantum measurements. A straightforward generalization based on the
noise-operator was used to reformulate Heisenberg's uncertainty relation for
the accuracy of simultaneous measurements to be universally valid and made the
conventional formulation testable to observe its violation. Recently, its
reliability was examined based on an anomaly that the error vanishes for some
imprecise measurements, in which the meter does not commute with the measured
observable. Here, we propose an improved definition for a quantum
generalization of the classical rms error, which is state-dependent,
operationally definable, and perfectly characterizes precise measurements.
Moreover, it is shown that the new notion maintains the previously obtained
universally valid uncertainty relations and their experimental confirmations
without changing their forms and interpretations, in contrast to a prevailing
view that a state-dependent formulation for measurement uncertainty relation is
not tenable.
