2005 Fiscal Year Final Research Report Summary
Statistical Estimation Theory for Quantum Channels
| Project/Area Number |
15340031
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Osaka University |
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Principal Investigator |
FUJIWARA Akio Osaka University, Graduate School of Science, Professor, 理学研究科, 教授 (30251359)
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| Co-Investigator(Kenkyū-buntansha) |
SUGIMOTO Mitsuru Osaka University, Graduate School of Science, Associate Professor, 理学研究科, 助教授 (60196756)
KONNO Kazuhiro Osaka University, Graduate School of Science, Professor, 理学研究科, 教授 (10186869)
SUZUKU Joe Osaka University, Graduate School of Science, Associate Professor, 理学研究科, 助教授 (50216397)
MABUCHI Toshiki Osaka University, Graduate School of Science, Professor, 理学研究科, 教授 (80116102)
WATANABE Takao Osaka University, Graduate School of Science, Professor, 理学研究科, 教授 (30201198)
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| Project Period (FY) |
2003 – 2005
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| Keywords | quantum channel / complete positivity / information geometry / adaptive estimation / Cramer-Rao inequality / martingale / representation theory / asymptotics |
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| Research Abstract |
This research addresses the problem of estimating an unknown quantum channel, based on noncommutative statistics and quantum information geometry. Main results are summarized as follows :
1. Estimation theory for generalized Pauli channels, generalized amplitude-damping channel, SU(d) channel, etc. :
These channels are the standard ones often used in quantum information theory. Based on noncommutative statistics, quantum information geometry, representation theory, as well as methods of experimental mathematics on a computer, we have obtained the optimal estimation scheme for each of those quantum channels. We also have clarified the underlining differential geometrical mechanism, in particular, the dilation/collapse of quantum statistical manifolds of output quantum states with respect to the quantum entanglement and the degree of extensions, behind the optimality of those estimation schemes.
2. Asymptotic theory of adaptive estimation schemes based on the locally unbiased estimators :
The notion of locally unbiased estimators, advocated first by Holevo, has been the target of criticism because the optimal estimation scheme depends on the unknown parameter itself. In order to surmount this difficulty, Nagaoka advocated an adaptive estimation scheme which make use of maximal likelihood estimators as a tentative estimator. However, due to the mathematical difficulty, the analysis of its asymptotic property has been left untouched. In this research, we have proved the strong consistency and the asymptotic efficiency of Nagaoka's adaptive estimation scheme based on the martingale theory.
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Research Products
(12 results)
