Information-disturbance trade-off in generalized entanglement swapping

We study information-disturbance trade-off in generalized entanglement swapping protocols wherein starting from Bell pairs $\left(1,2\right)$ and $\left(3,4\right)$, one performs an arbitrary joint measurement on $\left(2,3\right)$, so that $\left(1,4\right)$ now becomes correlated. We obtain trade-off inequalities between information gain in correlations of $\left(1,4\right)$ and residual information in correlations of $\left(1,2\right)$ and $\left(3,4\right)$ 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 $\left(2,3\right)$ can conserve information, even if, as in our example, the post-measurement states of all three pairs $\left(1,2\right)$, $\left(3,4\right)$, and $\left(1,4\right)$ 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.

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