dorsal/arxiv
View SchemaQuantization and noiseless measurements
| Authors | J. Kiukas, P. Lahti |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0612054 |
| URL | https://arxiv.org/abs/quant-ph/0612054 |
| DOI | 10.1088/1751-8113/40/9/014 |
| Journal | J. Phys. A: Math. Theor. 40 (2007) 2083-2091 |
Abstract
In accordance with the fact that quantum measurements are described in terms of positive operator measures (POMs), we consider certain aspects of a quantization scheme in which a classical variable $f:\R^2\to \R$ is associated with a unique positive operator measure (POM) $E^f$, which is not necessarily projection valued. The motivation for such a scheme comes from the well-known fact that due to the noise in a quantum measurement, the resulting outcome distribution is given by a POM and cannot, in general, be described in terms of a traditional observable, a selfadjoint operator. Accordingly, we notice that the noiseless measurements are the ones which are determined by a selfadjoint operator. The POM $E^f$ in our quantization is defined through its moment operators, which are required to be of the form $\Gamma(f^k)$, $k\in \N$, with $\Gamma$ a fixed map from classical variables to Hilbert space operators. In particular, we consider the quantization of classical \emph{questions}, that is, functions $f:\R^2\to\R$ taking only values 0 and 1. We compare two concrete realizations of the map $\Gamma$ in view of their ability to produce noiseless measurements: one being the Weyl map, and the other defined by using phase space probability distributions.
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"abstract": "In accordance with the fact that quantum measurements are described in terms\nof positive operator measures (POMs), we consider certain aspects of a\nquantization scheme in which a classical variable $f:\\R^2\\to \\R$ is associated\nwith a unique positive operator measure (POM) $E^f$, which is not necessarily\nprojection valued. The motivation for such a scheme comes from the well-known\nfact that due to the noise in a quantum measurement, the resulting outcome\ndistribution is given by a POM and cannot, in general, be described in terms of\na traditional observable, a selfadjoint operator. Accordingly, we notice that\nthe noiseless measurements are the ones which are determined by a selfadjoint\noperator. The POM $E^f$ in our quantization is defined through its moment\noperators, which are required to be of the form $\\Gamma(f^k)$, $k\\in \\N$, with\n$\\Gamma$ a fixed map from classical variables to Hilbert space operators. In\nparticular, we consider the quantization of classical \\emph{questions}, that\nis, functions $f:\\R^2\\to\\R$ taking only values 0 and 1. We compare two concrete\nrealizations of the map $\\Gamma$ in view of their ability to produce noiseless\nmeasurements: one being the Weyl map, and the other defined by using phase\nspace probability distributions.",
"arxiv_id": "quant-ph/0612054",
"authors": [
"J. Kiukas",
"P. Lahti"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/1751-8113/40/9/014",
"journal_ref": "J. Phys. A: Math. Theor. 40 (2007) 2083-2091",
"title": "Quantization and noiseless measurements",
"url": "https://arxiv.org/abs/quant-ph/0612054"
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