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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

Why this formula appears here

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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ρrad\rho_{\mathrm{rad}}

Symbol rho_rad

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

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Published contexts (1)

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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,

Equation 51 · Evolutionary Physics

The Bit Comes Back Before the Bearing

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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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