= Identical-resource entanglement swapping
{title2=$|\chi\rangle\propto\sum_rb_r^2|\xi_r\xi_r\rangle$}
Take two identical normalized entangled qubit pairs $\sum_{r=0}^1b_r|\xi_r\eta_r\rangle$. Project their two $\eta$ systems onto $|\Phi\rangle=2^{-1/2}\sum_k|\eta_k\eta_k\rangle$. The overlap with $|\eta_r\eta_s\rangle$ is $\delta_{rs}/\sqrt2$, so the unnormalized remote state is $\sum_rb_r^2|\xi_r\xi_r\rangle/\sqrt2$. Its squared norm gives success probability $\tfrac12\sum_r|b_r|^4$, and its normalized Schmidt coefficients are $b_r^2/(\sum_s|b_s|^4)^{1/2}$. At least two survive whenever the original pair was entangled. This is a conditional <entanglement swapping> result; disregarding the local measurement outcome leaves the remote reduced state unchanged. The coefficients are squared amplitudes, not their absolute squares, because the two input kets multiply before the Bell-state contraction.
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