 Identical-resource entanglement swapping

Take two identical normalized entangled qubit pairs ${\sum }_{r=0}^1{b}_r|{\xi }_r{\eta }_r⟩$. Project their two $\eta$ systems onto $|\Phi ⟩=2^{-1/2}{\sum }_k|{\eta }_k{\eta }_k⟩$. The overlap with $|{\eta }_r{\eta }_s⟩$ is ${\delta }_{rs}/\sqrt{2}$, so the unnormalized remote state is ${\sum }_r{b}_r^2|{\xi }_r{\xi }_r⟩/\sqrt{2}$. Its squared norm gives success probability $\frac{1}{2}{\sum }_r|b_r{|}^4$, 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.