  The principle of complementarity is quantified in two ways: by a universal
uncertainty relation valid for arbitrary joint estimates of any two observables
from a given measurement setup, and by a general uncertainty relation valid for
the_optimal_ estimates of the same two observables when the state of the system
prior to measurement is known. A formula is given for the optimal estimate of
any given observable, based on arbitrary measurement data and prior information
about the state of the system, which generalises and provides a more robust
interpretation of previous formulas for ``local expectations'' and ``weak
values'' of quantum observables. As an example, the canonical joint measurement
of position X and momentum P corresponds to measuring the commuting operators
X_J=X+X', P_J=P-P', where the primed variables refer to an auxilary system in a
minimum-uncertainty state. It is well known that Delta X_J Delta P_J >= hbar.
Here it is shown that given the_same_ physical experimental setup, and
knowledge of the system density operator prior to measurement, one can make
improved joint estimates X_est and P_est of X and P. These improved estimates
are not only statistically closer to X and P: they satisfy Delta X_est Delta
P_est >= hbar/4, where equality can be achieved in certain cases. Thus one can
do up to four times better than the standard lower bound (where the latter
corresponds to the limit of_no_ prior information). Other applications include
the heterodyne detection of orthogonal quadratures of a single-mode optical
field, and joint measurements based on Einstein-Podolsky-Rosen correlations.
