dorsal/arxiv
View SchemaQuantum States Estimation: Root Approach
| Authors | Yu. I. Bogdanov |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0310011 |
| URL | https://arxiv.org/abs/quant-ph/0310011 |
| DOI | 10.1117/12.562729 |
Abstract
Multiparametric statistical model providing stable reconstruction of parameters by observations is considered. The only general method of this kind is the root model based on the representation of the probability density as a squared absolute value of a certain function, which is referred to as a psi-function in analogy with quantum mechanics. The psi-function is represented by an expansion in terms of an orthonormal set of functions. It is shown that the introduction of the psi-function allows one to represent the Fisher information matrix as well as statistical properties of the estimator of the state vector (state estimator) in simple analytical forms. A new statistical characteristic, a confidence cone, is introduced instead of a standard confidence interval. The chi-square test is considered to test the hypotheses that the estimated vector converges to the state vector of a general population and that both samples are homogeneous. The expansion coefficients are estimated by the maximum likelihood method. The method proposed may be applied to its full extent to solve the statistical inverse problem of quantum mechanics (root estimator of quantum states). In order to provide statistical completeness of the analysis, it is necessary to perform measurements in mutually complementing experiments (according to the Bohr terminology). The maximum likelihood technique and likelihood equation are generalized in order to analyze quantum mechanical experiments. It is shown that the requirement for the expansion to be of a root kind can be considered as a quantization condition making it possible to choose systems described by quantum mechanics from all statistical models consistent, on average, with the laws of classical mechanics.
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"abstract": "Multiparametric statistical model providing stable reconstruction of\nparameters by observations is considered. The only general method of this kind\nis the root model based on the representation of the probability density as a\nsquared absolute value of a certain function, which is referred to as a\npsi-function in analogy with quantum mechanics. The psi-function is represented\nby an expansion in terms of an orthonormal set of functions. It is shown that\nthe introduction of the psi-function allows one to represent the Fisher\ninformation matrix as well as statistical properties of the estimator of the\nstate vector (state estimator) in simple analytical forms. A new statistical\ncharacteristic, a confidence cone, is introduced instead of a standard\nconfidence interval. The chi-square test is considered to test the hypotheses\nthat the estimated vector converges to the state vector of a general population\nand that both samples are homogeneous. The expansion coefficients are estimated\nby the maximum likelihood method. The method proposed may be applied to its\nfull extent to solve the statistical inverse problem of quantum mechanics (root\nestimator of quantum states). In order to provide statistical completeness of\nthe analysis, it is necessary to perform measurements in mutually complementing\nexperiments (according to the Bohr terminology). The maximum likelihood\ntechnique and likelihood equation are generalized in order to analyze quantum\nmechanical experiments. It is shown that the requirement for the expansion to\nbe of a root kind can be considered as a quantization condition making it\npossible to choose systems described by quantum mechanics from all statistical\nmodels consistent, on average, with the laws of classical mechanics.",
"arxiv_id": "quant-ph/0310011",
"authors": [
"Yu. I. Bogdanov"
],
"categories": [
"quant-ph"
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"doi": "10.1117/12.562729",
"title": "Quantum States Estimation: Root Approach",
"url": "https://arxiv.org/abs/quant-ph/0310011"
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