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Comparison Between the Cramer-Rao and the Mini-max Approaches in Quantum Channel Estimation

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Abstract

In a unified viewpoint in quantum channel estimation, we compare the Cramér-Rao and the mini-max approaches, which gives the Bayesian bound in the group covariant model. For this purpose, we introduce the local asymptotic mini-max bound, whose maximum is shown to be equal to the asymptotic limit of the mini-max bound. It is shown that the local asymptotic mini-max bound is strictly larger than the Cramér-Rao bound in the phase estimation case while both bounds coincide when the minimum mean square error decreases with the order \({O(\frac{1}{n})}\) . We also derive a sufficient condition so that the minimum mean square error decreases with the order \({O(\frac{1}{n})}\) .

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Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

    Masahito Hayashi

  2. Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore, 117542, Singapore

    Masahito Hayashi

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  1. Masahito Hayashi
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Correspondence to Masahito Hayashi.

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Communicated by M.B. Ruskai

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Hayashi, M. Comparison Between the Cramer-Rao and the Mini-max Approaches in Quantum Channel Estimation. Commun. Math. Phys. 304, 689–709 (2011). https://doi.org/10.1007/s00220-011-1239-4

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  • DOI: https://doi.org/10.1007/s00220-011-1239-4

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