  We argue for an operational requirement that all state-dependent measures of
disturbance should satisfy. Motivated by this natural criterion, we prove that
in any $d$-dimensional Hilbert space and for any pair of non-commuting
operators, $A$ and $B$, there exists a set of at least $2^{d-1}$ zero-noise,
zero-disturbance (ZNZD) states, for which the first observable can be measured
without noise and the second will not be disturbed. Moreover, we show that it
is possible to construct such ZNZD states for which the expectation value of
the commutator $[A,B]$ does not vanish. Therefore any state-dependent
error-disturbance relation, based on the expectation value of the commutator as
a lower bound, must violate the operational requirement. We also discuss
Ozawa's state-dependent error-disturbance relation in light of our results and
show that the disturbance measure used in this relation exhibits unphysical
properties. We conclude that the trade-off is inevitable only between
state-independent measures of error and disturbance.
