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Equation 51 · Part 1 · The Bit Comes Back Before the Bearing

Symbol rho_rad

ρrad(g)=Urad(g) ρrad(0) Urad(g)†,\rho_{\mathrm{rad}}(g) = U_{\mathrm{rad}}(g)\,\rho_{\mathrm{rad}}(0)\,U_{\mathrm{rad}}(g)^\dagger,
ρrad\rho_{\mathrm{rad}}

What this part means

rhoro_rad is part of the quantity the equation computes from the expression on the right.

Its job in the formula

rhoro_rad is part of the quantity the equation computes from the expression on the right.

The passage around this formula

The covariance equation from the previous section is the formal cost of that fact. Because V commutes with the group action, the radiation’s reduced state along the true heading is a rotated copy of the same state at heading zero, ρrad(g)=Urad(g) ρrad(0) Urad(g)†\rho_{\mathrm{rad}}(g) = U_{\mathrm{rad}}(g)\,\rho_{\mathrm{rad}}(0)\,U_{\mathrm{rad}}(g)^\dagger. which is a genuine constraint, not a simplifying choice: FQF_Q[ρrad(g)\rho_{\mathrm{rad}}(g)] is therefore identical at every point along this orbit, since the quantum Fisher information depends only on the local geometry of the state family under the generator, and a rotated copy of the same family has the same local geometry everywhere. tG(δ)t_G(\delta) does not depend on which heading is being estimated. That is the covariance check this construction owes…

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Sources cited in the article section

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