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

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 generalizes 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 $X_J=X+X^{\prime }$ and $P_J=P-P^{\prime }$, where the primed variables refer to an auxiliary system in a minimum-uncertainty state. It is well known that $\Delta X_J\Delta P_J⩾\hslash$. 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 $X_{opt}$ and $P_{opt}$ of $X$ and $P$. These improved estimates are not only statistically closer to $X$ and $P$ but further satisfy $\Delta X_{opt}\Delta P_{opt}⩾\hslash ∕4$, 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.