= Average pure-source fidelity after typical projection
{title2=$F\geq\sum_kp_k\alpha_k^4\geq2\operatorname{Tr}(\rho P)-1$}
For a <quantum state ensemble> of pure vectors and an <orthogonal projection> $P$, measure $\{P,I-P\}$, preserve the successful subspace, and decode failure to a fixed normalized vector $\varphi$. The resulting <quantum channel> is $\Lambda(X)=PXP+\operatorname{Tr}[(I-P)X]|\varphi\rangle\langle\varphi|$. For $\alpha_k=\|P\psi_k\|$, its <squared quantum fidelity> on the $k$th source vector is $\alpha_k^4+(1-\alpha_k^2)|\langle\psi_k|\varphi\rangle|^2$. The inequality $a^2\geq2a-1$ on $[0,1]$ proves the displayed average bound. It depends only on the source density operator's weight in $P$ and does not require the emitted vectors to be orthogonal.
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