dorsal/arxiv
View SchemaGeneral Probabilistic Framework of Randomness
| Authors | Elena R. Loubenets |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0305126 |
| URL | https://arxiv.org/abs/quant-ph/0305126 |
Abstract
We introduce a new mathematical framework for the probabilistic description of an experiment upon a system of any type in terms of initial information representing this system. Based on the notions of an information state, an information state space and a generalized observable, this general framework covers the description of a wide range of experimental situations including those where, with respect to a system, an experiment is perturbing. We prove that, to any experiment upon a system, there corresponds a unique generalized observable on a system initial information state space, which defines the probability distribution of outcomes under this experiment. We specify the case where initial information on a system provides "no knowledge" for the description of an experiment. Incorporating in a uniform way the basic notions of conventional probability theory and the non-commutativity aspects and the basic notions of quantum measurement theory, our framework clarifies the principle difference between Kolmogorov's model in probability theory and the statistical model of quantum theory. Both models are included into our framework as particular cases. We show that the phenomenon of "reduction" of a system initial information state is inherent, in general, to any non-destructive experiment and upon a system of any type. Based on our general framework, we introduce the probabilistic model for the description of non-destructive experiments upon a quantum system and prove that positive bounded linear mappings on the Banach space of trace class operators, arising in the description of experiments upon a quantum system, are completely positive.
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"abstract": "We introduce a new mathematical framework for the probabilistic description\nof an experiment upon a system of any type in terms of initial information\nrepresenting this system. Based on the notions of an information state, an\ninformation state space and a generalized observable, this general framework\ncovers the description of a wide range of experimental situations including\nthose where, with respect to a system, an experiment is perturbing. We prove\nthat, to any experiment upon a system, there corresponds a unique generalized\nobservable on a system initial information state space, which defines the\nprobability distribution of outcomes under this experiment. We specify the case\nwhere initial information on a system provides \"no knowledge\" for the\ndescription of an experiment. Incorporating in a uniform way the basic notions\nof conventional probability theory and the non-commutativity aspects and the\nbasic notions of quantum measurement theory, our framework clarifies the\nprinciple difference between Kolmogorov\u0027s model in probability theory and the\nstatistical model of quantum theory. Both models are included into our\nframework as particular cases. We show that the phenomenon of \"reduction\" of a\nsystem initial information state is inherent, in general, to any\nnon-destructive experiment and upon a system of any type. Based on our general\nframework, we introduce the probabilistic model for the description of\nnon-destructive experiments upon a quantum system and prove that positive\nbounded linear mappings on the Banach space of trace class operators, arising\nin the description of experiments upon a quantum system, are completely\npositive.",
"arxiv_id": "quant-ph/0305126",
"authors": [
"Elena R. Loubenets"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP",
"math.PR"
],
"title": "General Probabilistic Framework of Randomness",
"url": "https://arxiv.org/abs/quant-ph/0305126"
},
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