Heisenberg's uncertainty principle has been understood to set a limitation on measurements, whereas the long
standing mathematical formulation does not allow such an interpretation. Recently, a new relation was found
by the present author to give a universally valid relation between noise and disturbance in general quantum
measurements, and it has become clear that the new relation plays a role of the first principle to derive various
quantum limits on measurement and information processing. Here, we discuss the state-dependent notion of
precise measurements of a given observable, and consider a perfect distance such that zero distance implies the
perfect correlation. Then, we shall show that even in the perfect error notions both the position measuring noise
and the momentum disturbance can be arbitrarily small. We also show that it is possible to generalize the new
noise-disturbance uncertainty relation to perfect error distances.
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