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The principle of complementarity is quantified in two ways: by a universaluncertainty relation valid for arbitrary joint estimates of any two observablesfrom a given measurement setup, and by a general uncertainty relation valid forthe_optimal_ estimates of the same two observables when the state of the systemprior to measurement is known. A formula is given for the optimal estimate ofany given observable, based on arbitrary measurement data and prior informationabout the state of the system, which generalises and provides a more robustinterpretation of previous formulas for ``local expectations'' and ``weakvalues'' of quantum observables. As an example, the canonical joint measurementof position X and momentum P corresponds to measuring the commuting operatorsX_J=X+X', P_J=P-P', where the primed variables refer to an auxilary system in aminimum-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, andknowledge of the system density operator prior to measurement, one can makeimproved joint estimates X_est and P_est of X and P. These improved estimatesare not only statistically closer to X and P: they satisfy Delta X_est DeltaP_est >= hbar/4, where equality can be achieved in certain cases. Thus one cando up to four times better than the standard lower bound (where the lattercorresponds to the limit of_no_ prior information). Other applications includethe heterodyne detection of orthogonal quadratures of a single-mode opticalfield, and joint measurements based on Einstein-Podolsky-Rosen correlations.Comment: revtex4, no figures, much improved exposition of optimal estimates  (thanks to excellent referee), to appear in PR

Prior information: How to circumvent the standard joint-measurement uncertainty relation