Information-disturbance trade-off in generalized entanglement swapping

We study information-disturbance trade-off in generalized entanglement swapping protocols wherein starting from Bell pairs left\(1,2right\) and left\(3,4right\), one performs an arbitrary joint measurement on left\(2,3right\), so that left\(1,4right\) now becomes correlated. We obtain trade-off inequalities between information gain in correlations of left\(1,4right\) and residual information in correlations of left\(1,2right\) and left\(3,4right\) respectively and argue that information contained in correlations (information) is conserved if each inequality is an equality. We show that information is conserved for a maximally entangled measurement but is not conserved for any other complete orthogonal measurement and Bell measurement mixed with white noise. However, rather surprisingly, we find that information is conserved for rank-two Bell diagonal measurements, although such measurements do not conserve entanglement. We also show that a separable measurement on left\(2,3right\) can conserve information, even if, as in our example, the post-measurement states of all three pairs left\(1,2right\), left\(3,4right\), and left\(1,4right\) become separable. This implies correlations from an entangled pair can be transferred to separable pairs in nontrivial ways so that no information is lost in the process.