  The notions of error and disturbance appearing in quantum uncertainty
relations are often quantified by the discrepancy of a physical quantity from
its ideal value. However, these real and ideal values are not the outcomes of
simultaneous measurements, and comparing the values of unmeasured observables
is not necessarily meaningful according to quantum theory. To overcome these
conceptual difficulties, we take a different approach and define error and
disturbance in an operational manner. In particular, we formulate both in terms
of the probability that one can successfully distinguish the actual measurement
device from the relevant hypothetical ideal by any experimental test
whatsoever. This definition itself does not rely on the formalism of quantum
theory, avoiding many of the conceptual difficulties of usual definitions. We
then derive new Heisenberg-type uncertainty relations for both joint
measurability and the error-disturbance tradeoff for arbitrary observables of
finite-dimensional systems, as well as for the case of position and momentum.
Our relations may be directly applied in information processing settings, for
example to infer that devices which can faithfully transmit information
regarding one observable do not leak any information about conjugate
observables to the environment. We also show that Englert's wave-particle
duality relation [PRL 77, 2154 (1996)] can be viewed as an error-disturbance
uncertainty relation.
