$$\operatorname{Var}\hat g \;\ge\; \frac{1}{F_Q\big[\rho_{\mathrm{rad}}(g)\big]}, \qquad t_G(\delta) \;=\; \min\Big\{\,t : \operatorname{Var}\hat g\big|_{R\cup C_t} \le \delta^2\,\Big\}$$

Equation 37 · Evolutionary Physics

The Bit Comes Back Before the Bearing

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

For the compass, returned has to mean something else, because g is 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$\cup$ $C_t$ , applies the best available measurement and estimator $\hat g$ , and the achievable mean-squared error is bounded below by the quantum Cramér-Rao bound, $\operatorname{Var}\hat g \;\ge\; \frac{1}{F_Q\big[\rho_{\mathrm{rad}}(g)\big]}, \qquad t_G(\delta) \;=\; \min\Big\{\,t : \operatorname{Var}\hat g\big|_{R\cup C_t} \le \delta^2\,\Big\}$. $F_Q$ is the quantum Fisher information of the radiation’s reduced state with respect to g ; it carries units of $g^{-2}$ , i.e. inverse radians squared, and since g is already dimensionless, $F_Q$ is a pure number. $\delta$ is the alignment tolerance the observer has decided to demand, also in…

Meanings in this article

• $\delta$: the alignment tolerance the observer has decided to demand, also in radians, and $t_G(\delta)$ has units of time [ 12 ].
• $\delta^2$: the square of delta; the alignment tolerance the observer has decided to demand, also in radians, and $t_G(\delta)$ has units of time [ 12 ].

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