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Equation 37 · Part 1 · The Bit Comes Back Before the Bearing

Symbol hat g

Var⁡g^  ≥  1FQ[ρrad(g)],tG(δ)  =  min⁡{ t:Var⁡g^∣R∪Ct≤δ2 }.\operatorname{Var}\hat g \;\ge\; \frac{1}{F_Q\big[\rho_{\mathrm{rad}}(g)\big]}, \qquad t_G(\delta) \;=\; \min\Big\{\,t : \operatorname{Var}\hat g\big|_{R\cup C_t} \le \delta^2\,\Big\}.
g^\hat g

What this part means

hat g is the quantity selected or evaluated by the optimization written on the right.

Its job in the formula

hat g is the quantity selected or evaluated by the optimization written on the right.

The passage around this formula

…a continuous parameter and no channel can reproduce an unknown continuous value exactly. The honest promise is metrological: the same agent, using the same R∪\cup CtC_t , applies the best available measurement and estimator g^\hat g , and the achievable mean-squared error is bounded below by the quantum Cramér-Rao bound, Var⁡g^  ≥  1FQ[ρrad(g)],tG(δ)  =  min⁡{ t:Var⁡g^∣R∪Ct≤δ2 }\operatorname{Var}\hat g \;\ge\; \frac{1}{F_Q\big[\rho_{\mathrm{rad}}(g)\big]}, \qquad t_G(\delta) \;=\; \min\Big\{\,t : \operatorname{Var}\hat g\big|_{R\cup C_t} \le \delta^2\,\Big\}. FQF_Q is the quantum Fisher information of the radiation’s reduced state with respect to g ; it carries units of g−2g^{-2} , i.e. inverse radians squared, and since g is already…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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