2003/09/30 by Michael J. W. Hall · 5 citations
Physics and Astronomy · #quant-ph
paper · pdf · doi:10.1103/physreva.69.052113
published as Phys. Rev. A 69 (2004) 052113 · revtex4, no figures, much improved exposition of optimal estimates (thanks to excellent referee), to appear in PRA
arxiv created 2004/03/30 · arxiv updated 2009/12/01
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 theoptimal_ 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 XJ=X+X', PJ=P-P', where the primed variables refer to an auxilary system in a minimum-uncertainty state. It is well known that Delta XJ Delta PJ >= hbar. Here it is shown that given thesame_ physical experimental setup, and knowledge of the system density operator prior to measurement, one can make improved joint estimates Xest and Pest of X and P. These improved estimates are not only statistically closer to X and P: they satisfy Delta Xest Delta Pest >= 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 ofno_ 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.