  It is shown that the well-defined unbiased measurement or disturbance of a
dynamical variable is not maintained for the precise measurement of the
conjugate variable, independently of uncertainty relations. The conditionally
valid uncertainty relations on the basis of those additional assumptions, which
include most of the familiar Heisenberg-type relations, thus become singular
for the precise measurement. We clarify some contradicting conclusions in the
literature concerning those conditionally valid uncertainty relations: The
failure of a naive Heisenberg-type error-disturbance relation and the modified
Arthurs-Kelly relation in the recent spin measurement is attributed to this
singular behavior. The naive Heisenberg-type error-disturbance relation is
formally preserved in quantum estimation theory, which is shown to be based on
the strict unbiased measurement and disturbance, but it leads to unbounded
disturbance for bounded operators such as spin variables. In contrast, the
Heisenberg-type error-error uncertainty relation and the Arthurs-Kelly
relation, as conditionally valid uncertainty relations, are consistently
maintained.
