dorsal/arxiv
View SchemaComparative Noninformativities of Quantum Priors Based on Monotone Metrics
| Authors | Paul B. Slater |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9703012 |
| URL | https://arxiv.org/abs/quant-ph/9703012 |
| DOI | 10.1016/S0375-9601(98)00556-8 |
| Journal | Phys.Lett. A247 (1998) 1-8 |
Abstract
We consider a family of prior probability distributions of particular interest, all being defined on the three-dimensional convex set of two-level quantum systems. Each distribution is, following recent work of Petz and Sudar, taken to be proportional to the volume element of a monotone metric on that Riemannian manifold. We apply an entropy-based test (a variant of one recently developed by Clarke) to determine which of two priors is more noninformative in nature. This involves converting them to posterior probability distributions based on some set of hypothesized outcomes of measurements of the quantum system in question. It is, then, ascertained whether or not the original relative entropy (Kullback-Leibler distance) between a pair of priors increases or decreases when one of them is exchanged with its corresponding posterior. The findings lead us to assert that the maximal monotone metric yields the most noninformative (prior) distribution and the minimal monotone (that is, the Bures) metric, the least. Our conclusions both agree and disagree, in certain respects, with ones recently reached by Hall, who relied upon a less specific test criterion than our entropy-based one.
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"abstract": "We consider a family of prior probability distributions of particular\ninterest, all being defined on the three-dimensional convex set of two-level\nquantum systems. Each distribution is, following recent work of Petz and Sudar,\ntaken to be proportional to the volume element of a monotone metric on that\nRiemannian manifold. We apply an entropy-based test (a variant of one recently\ndeveloped by Clarke) to determine which of two priors is more noninformative in\nnature. This involves converting them to posterior probability distributions\nbased on some set of hypothesized outcomes of measurements of the quantum\nsystem in question. It is, then, ascertained whether or not the original\nrelative entropy (Kullback-Leibler distance) between a pair of priors increases\nor decreases when one of them is exchanged with its corresponding posterior.\nThe findings lead us to assert that the maximal monotone metric yields the most\nnoninformative (prior) distribution and the minimal monotone (that is, the\nBures) metric, the least. Our conclusions both agree and disagree, in certain\nrespects, with ones recently reached by Hall, who relied upon a less specific\ntest criterion than our entropy-based one.",
"arxiv_id": "quant-ph/9703012",
"authors": [
"Paul B. Slater"
],
"categories": [
"quant-ph",
"physics.data-an"
],
"doi": "10.1016/S0375-9601(98)00556-8",
"journal_ref": "Phys.Lett. A247 (1998) 1-8",
"title": "Comparative Noninformativities of Quantum Priors Based on Monotone Metrics",
"url": "https://arxiv.org/abs/quant-ph/9703012"
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