dorsal/arxiv
View SchemaPrior information: how to circumvent the standard joint-measurement uncertainty relation
| Authors | Michael J. W. Hall |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0309091 |
| URL | https://arxiv.org/abs/quant-ph/0309091 |
| DOI | 10.1103/PhysRevA.69.052113 |
| Journal | Phys. Rev. A 69 (2004) 052113 |
Abstract
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.
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"abstract": "The principle of complementarity is quantified in two ways: by a universal\nuncertainty relation valid for arbitrary joint estimates of any two observables\nfrom a given measurement setup, and by a general uncertainty relation valid for\nthe_optimal_ estimates of the same two observables when the state of the system\nprior to measurement is known. A formula is given for the optimal estimate of\nany given observable, based on arbitrary measurement data and prior information\nabout the state of the system, which generalises and provides a more robust\ninterpretation of previous formulas for ``local expectations\u0027\u0027 and ``weak\nvalues\u0027\u0027 of quantum observables. As an example, the canonical joint measurement\nof position X and momentum P corresponds to measuring the commuting operators\nX_J=X+X\u0027, P_J=P-P\u0027, where the primed variables refer to an auxilary system in a\nminimum-uncertainty state. It is well known that Delta X_J Delta P_J \u003e= hbar.\nHere it is shown that given the_same_ physical experimental setup, and\nknowledge of the system density operator prior to measurement, one can make\nimproved joint estimates X_est and P_est of X and P. These improved estimates\nare not only statistically closer to X and P: they satisfy Delta X_est Delta\nP_est \u003e= hbar/4, where equality can be achieved in certain cases. Thus one can\ndo up to four times better than the standard lower bound (where the latter\ncorresponds to the limit of_no_ prior information). Other applications include\nthe heterodyne detection of orthogonal quadratures of a single-mode optical\nfield, and joint measurements based on Einstein-Podolsky-Rosen correlations.",
"arxiv_id": "quant-ph/0309091",
"authors": [
"Michael J. W. Hall"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.69.052113",
"journal_ref": "Phys. Rev. A 69 (2004) 052113",
"title": "Prior information: how to circumvent the standard joint-measurement uncertainty relation",
"url": "https://arxiv.org/abs/quant-ph/0309091"
},
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