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In the study of quantum limits to parameter estimation, the high dimensionality of the density operator and that of the unknown parameters have long been two of the most difficult challenges.
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More precisely, \mathaccentV t i l d e 07 E β \mathaccentV{tilde}07E\beta is the unique Riesz-Fréchet representation Reed and Simon 1980 of D h β D_{h}\beta as a continuous linear functional of h ∈ 𝒴 h\in\mathcal{Y} , δ \delta is that for h ∈ 𝒵 ⊂ 𝒴 h\in\mathcal{Z}\subset\mathcal{Y} , and δ eff \delta_{\rm eff} is that for h ∈ 𝒯 ⊆ 𝒵 ⊂ 𝒴 h\in\mathcal{T}\subseteq\mathcal{Z}\subset\mathcal{Y} Reed and Simon 1980 ; Bickel et al. 1993 . The existence of each relies on D h β D_{h}\beta being continuous with respect to h h in each domain, so the existence of \mathaccentV t i l d e 07 E β \mathaccentV{tilde}07E\beta implies that of δ \delta and δ eff \delta_{\rm eff}
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2007
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2016
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2016
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2016
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2017
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2017
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Refs. Tsang et al. 2016 ; Tsang 2017 ; Tsang 2019c use the symbol L L for the number of detected photons, which is stochastic, and N N for the expected number of detected photons. For optics models with Poisson statistics, the conditioning on the detected photon number does not introduce any significant difference to the theory
2017
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S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, “Advances in photonic quantum sensing,” Nature Photonics 12
2018
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2018
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2018
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2018
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2018
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I. Ruo Berchera and I. P. Degiovanni, “Quantum imaging with sub-Poissonian light: challenges and perspectives in optical metrology,” Metrologia 56
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